Kill Colonel House. Wilson's most trusted and least reliable advisor.
In general, I don't think it's that difficult, really. Saving Wilson's reputation, not killing House. The main, obvious thing would be to get the Treaty of Versailles ratified, along with whatever improvements in the negotiation process would have been required to get us there.
The other possibility, I think, would be for Wilson to be able to successfully negotiate an end to World War I in 1916, which included a League of Nations that was, on the whole, successful. This isn't enormously far from OTL: the US had both Germany and Britain at least interested in coming to terms, and it had the leverage needed via American lending and commerce - especially to the UK.
Needless to say, it didn't happen, but it wasn't enormously far away. Wilson and the US State department mainly lacked the sophistication to actually get the job done, and by year's end, Lloyd George had ousted Asquith and the opportunity had passed.
From there, though, you'd depend quite a bit on what actually came of the Wilson peace. The political will domestically would need some kind of thaw between Wilson and Roosevelt/Taft ( the latter of whom headed up an organization called "The League to Enforce Peace" during WWI). If the war ends in 1916 with the League coming to the table, there's probably enough political will to get it in place. You also have a Democratic Senate (56-40 going into the 1916 election, 54-42 coming out).
The peace itself in Europe is probably relatively straightforward, without major political shifts on the Western Front, and Germany making some kind of restitution to Belgium. The eastern front is more complicated; Wilson's peace efforts never really reached Russia, but any kind of peace in 1916 probably at least gets some kind of stability for the Romanovs.
It's the next few generations that really matter, at least for Wilson's reputation: will the League succeed in ushering in an era of peace and prosperity, along with American leadership? Or will it fall apart by 1930?