Whatever they might have been when the Russian Social Democrats split along pro or anti Lenin lines, by 1917 I am unaware of any Menshevik political organization whatsoever. Basically by that date I'd define a "Menshevik" as a Russian social democrat who followed Marx but not Lenin, and there was no working-class organization along these lines to give the intellectuals who fell into that category any traction. I've read some of the writings of one of them during the Revolutionary period and my impression is they were all intelligentsia, and typically managed to hold jobs working for the Tsarist administration.
Even Kerensky, insofar as he could claim any mass base at all, was a some-time functionary and agent of the Social Revolutionaries, who weren't even Marxist; they were an agrarian affiliation. They certainly did have a mass base, of peasants rather than industrial workers, but Kerensky's claims to power were more a reflection of his place in the "Provisional Government" then the SRs promoting him within their own ranks. The PG of course was pretty much a body of former Tsarist officials and notables, who appointed themselves and relied on no mass base; Kerensky was chosen by them as their nominal head in an attempt to co-opt the SRs, he in no way was imposed on them from below.
So the Mensheviks were not a party, they were a clique, and while individual ones doubtless had notions about how things ought to be run, there was no party program for them to rally support to as such. Their basic premise was that while Marxist laws of development held, these very laws decreed that backward, underdeveloped nations and peoples such as the Russian Empire would need to be transformed first into bourgeois capitalist nations, and only then would the ground work for a socialist proletariat be laid. It meant that the task at hand in Russia was to aid rather than oppose the rise of strictly capitalist and liberal power, with the current class enemy at hand being the pre-capitalist aristocracy. Whereas, given Russian backwardness, surely the pathfinders of post-capitalist socialist communism would be the working classes of the developed nations of Western Europe and perhaps the USA; if these places had not gone Red yet, surely it would be generations premature to work for such a revolution in Russia!
One can see it would be a very comfortable set of doctrines for a member of the Russian intelligentsia to hold; at the same time as they considered themselves the enlightened disciples of a new age of history, they could also enjoy the perks of privileged classes in the existing Russian order, ushering in a well-exampled edition of progress that itself would hold benign niches for the enlightened, while letting the French, Germans, British and Americans first stumble upon the painful land-mines of proletarian revolution and by trial and error, show a less traumatic way to poor Russia
One can also see why people like the Bolsheviks despised them. I don't know what the Social Revolutionaries thought of them; probably they paid little attention, their eyes fixed on players who had real power.
So, if you ask, what would they do if they had power, that is a lot like asking what I'd do if I commanded an omnipotent genie with unlimited wishes. The difficulty in figuring out how they'd get power (on their own terms, and not merely as agents of some other power base) isn't a matter of details or strategy, it is a matter of there being no engines or instruments whatsoever other than moral suasion and arguments of enlightened self-interest for quite different people, to assist, advise and perhaps guide capitalists with quite different ideals and a real organization toward very different purposes.
The closest I can come to it is asking, "what if the Mensheviks were basically correct?" This has negative and positive aspects--for instance, if they were correct then it would have been impossible for Lenin and his followers to gain traction among the Russian working classes, for in some ATL theoretical sense they'd be somehow unready or unwilling to take on roles and tasks that in fact OTL they did. The SRs would also be incorrect and therefore in some way or another would fail to achieve the organization they did OTL. The working classes would perforce be driven into line, submitting as their Western counterparts had to bourgeois law and their function as value-producing, expendable laborers producing profit for a relative handful of capitalists. A middle class would grow and spread and provide a sound political base for liberal parties that would favor broader spreading of knowledge and gradual political enfranchisement of the masses. But not before these masses had first had their pre-capitalist bases of livelihood liquidated and replaced by a comprehensive system of private property in the means of production for private gain. The peasant Mirs, the "communes" of the old order, would somehow or other wither or be abolished, in their place would be private landownership, whether on a broadly distributed basis a la Jeffersonian ideals or far more likely, highly consolidated in relatively few hands, and the lands worked by hired wage-earning laborers for the profit of the land-owners, who would presumably by the magic of the marketplace hone agricultural practices at their discretion to maximize output. Population would shift off the land into rising industrial centers and eventually the industrial workers would far outnumber the country workers, but both would be essentially wage-earners. For quite a long time they could expect to live in a limbo of immiseration, cut off from traditional sources of subsistence but earning low wages while the system grew and accumulated wealth in relatively few hands--but a broadening middle class would exist and grow, in sufficient numbers and with sufficient hopes to loyally uphold the liberal regime and guard it both from reaction and premature revolution. With these inevitable and natural evolutions, the aggregate wealth of the Russian Empire would rise, its military capabilities rise accordingly, so the prospect of foreign conquest could be discounted.
In time, per Marx, the middle classes would start to again shrink, the contradictions of capitalism would become painfully felt, the new proletariat, now seasoned by generations of service as workers hired and fired at pleasure would transfer workplace organization toward disciplined partisanship and begin to foresee the seizure of means of production and reorganization under working class party dictatorship. But long before that distant day of glory could come to Russia, the workers of Western Europe and North America would already have come to their own boils, and carried out their own bloody revolutions, far away and overseas, and on one hand Russian workers would have an inspiring example to organize with and around, and on the other the ruling classes would see the writing on the wall and, those who did not panic in futile reaction, take measures to coopt and moderate the coming revolution. Perhaps reaction would take the upper hand and enter into die-hard resistance against the foreign contamination of socialism, surely as one of the great powers among the reactionaries, though I suppose a Menshevik would say no, we'll have been guiding things with sage advice and the liberal reformers will with our guidance prevail, and so Russia gradually, perhaps not last among nations but far from first, will transition by painless degrees toward socialism and eventually communism, coming in quietly through the back door.
Now if someone with such hopes were to confront the grim realities of 1917 they'd find these hopes much clouded and darkened with doubt. Unaccountably, both Bolsheviks and SRs commanded masses, moving with unforeseen intelligence and resolve, who would have none of this gradual process if they could help it. The middle classes the Mensheviks hoped would be emerging naturally, as particularly hard-working and intelligent former members of the working classes gained culture and a stake in an orderly and progressive oligarchy of property, but would mostly arise from former aristocrats induced by the obsolescence of their old role and the opportunities of investing old wealth into new profitable avenues, seemed oddly absent and where represented at all, outnumbered badly and lacking the confidence and direction their inevitable historic destiny should be energizing them with, and what strength they had seemed rather to rely on tradition and violent measures; instead of know-how acquired from hard-won success in competitive business and legions of acolyte clerks and smaller business partners in their wake, relying on foreign-hired managers and the inspectors sent by foreign investors with an anxious eye on their property. What organization might stand opposed to the irresponsible, premature yet somehow lean, mean and organized masses, relied much more on old-time repressive disciplinary violence of the old regime.
What would an optimistic, hopeful Menshevik make of this nightmare, and what ways out could they foresee? In the circumstances, some might re-evaluate their Marx, and belatedly switch over to the Bolsheviks or perhaps, discarding Marx, the SRs. Others might abandon Marx and progressivism with it and go over to the Constitutional Democrats, the Kadets (CDs, we'd spell and say it) or even the old-timer reactionaries.
What could a more stalwart Menshevik do, but rejecting the deluded and doomed radicals, appear hat in hand alongside those defecting in earnest to the Kadets or the generals, and offer their services to them in the hope that historical inevitability will reassert itself, and by and by, after a necessary period of heavy-handed discipline progress might resume?
The alternative of going over to the radical camps would be less attractive; the past might be on the side of the reactionaries but the future (for those who did not reevaluate their Marxism and find Lenin making more sense, or were impressed by the premises of the Social Revolutionaries) could not favor these distant harbingers of the true Revolution. Before the proletarians triumph, they must first prematurely attempt an unfounded victory, perhaps hold the barricades or even storm the Bastille, and stand for a time, short or long, as a glorious if deformed premonition of that distant day, only to inevitably stumble and fall, to be mowed down and harried by the triumphant Right. Not only would they be trying to flee onto an already sinking ship, the crew of that ship would hate and despise them on sight; having fanatically slipped under the spell of some ideological adventurer, and spat upon them when they seemed to have common sense on their side, how would they treat the Mensheviks now that their world has gone so unaccountably and irrationally dark?
The premise of the thread is that the Mensheviks are not fundamentally wrong and can somehow prevail; in this juncture the only lifeline for them to better days in the hidden future would be to take their place with other Whites now, and roll with whatever agendas they may wish to impose with their power. Presumably the Reds must crumple and fail, later if not sooner, and having blighted and delayed Russia from its natural, historically inevitable course, eventually under the parties of Order amid the ruins, progress will resume and slowly heal the empire, and bring Russia back to its destiny, and the advice of the stalwart Mensheviks will, after some years or decades, fall welcome on powerful ears again. Perhaps after the bloodletting of a civil war, the turmoils of half-baked schemes of radical communization will do the work of parliamentary Enclosures Acts or American land-grabs, and open the way to the privatization of the countryside and the rise of urban middle classes, and at long last the Mensheviki can begin to form an actual party and organization, and in the name of reformist progress take over the government and guide society.
But remember they still embody a contradiction; to be a Menshevik is to favor capitalism in the short run but also to look forward to the equally inevitable abolition of that same privatized utopia for another one that must again embody a social earthquake. To be Mensheviks openly and sincerely, they must proclaim that for now the task is to build a whole new order, but only as scaffolding to be torn apart again, in favor of something better that is currently impossible. But unless those who play the role of the pioneers and defenders of bourgeois liberalism today accept that their role and task, or that of their children or grandchildren, will change to something quite different that will leave them in a less relatively privileged state when it is ready, and induce enthusiasm and commitment to this two-stage plan, who would follow the Mensheviks? Would not the rising new capitalist/bourgeois powerhouses simply seek to retain power on their own terms forever, and despise and fear the second phase and anyone who favors it? If by subterfuge and guile, the Mensheviks instead hide their messianic dreams of the future proletarian revolution and focus solely on reforms in the present, they probably will forget their Marxism completely, and if they hew to the ultimate goal and merely excuse current capitalism as a necessary if painful phase, what support can they find anywhere? If Russia had not faced the crises of WWI and its aftermath, such a two-phase philosophy might have been sustainable as a fringe movement, and as the real Revolution approached at last, they would gradually become Bolsheviks in manner and belief and prospects. But having undergone Civil War to suppress premature Reds, surely the victorious Whites would be vigilant against any breath of Marxist heresy, and ruthlessly cut it down wherever it may be found?
I really can't envision any scenario then where the Menshevik vision can accrue a power basis on its own terms; insofar as it is correct, there is no need of organization or action to guarantee its triumph.
