If I define 'best' is having prevented the bulk of armed conflicts around the world, then Yellowstone should erupt in late 1914 or early 1915.
The knock on effects are fairly snowballing. With the ashclouds comming out in the winter of 1914, 1915 is a year without a summer.
However more immidiately the Panama-Pacific International Exposition never happens because of the ash clouds and in general the American west coast is sent into frenzy, and as such San Fransico never truely recovers from the 1906 quake and people leave the American West coast in droves.
By April, low temperatures in Europe have made poison gas ineffective on both the Russian front, and during the Battles around Ypres and Artois, causing high German losses, but poor weather also prevents successes in Turkey by the Allies, thus delaying Italys entrance into the Great War by at least a year. Futhermore later by August, the oil lubricants required for the 1st proto-type tank for the British aren't working as intended due to the cold and damp.
The British war committee is not impressed by the results and so futher development is scrapped.
As the year draws to a close, the lawlessness in the western united states allows the Mexican Revolution to spill over into the united states...or more propperly refugees seeking to remove themselves from ashclouds from continuing eruptions;
Flee south into the Mexican Revolution, stirring up futher sentiments in the region putting American-Mexican tensions high.
By Feb 1916 temperatures are at a record cold, and the Battle of Verdun is fought in the snow and ice, however because Italy didn't join the year before additional soliders and shells are avalible for the German spring offensive, (since the weather patterns are screwed up anyhow the 10 day blizzard that would have slowed the German advance doesn't have the 'misfortune' to happen at that time).
Because the war committee doesn't develop the 'tank' the previous year, the Somme offensive is less effective, and becomes known as the 'Bloody days of the Somme' with over 1.2 million casulities.
During the August, tensions with Mexico and America are brought to ahead with the Jones Act and Mexicos latent support for the pro-spanish stance following the 1898 Spanish-American war; the Mexicans beliving that the Americans are just 'giving away' rightfully spanih territories, and so an anti-american, pro-New Spain movement begins to shape the revolution, much fuelled by Americans trying to 'carve out a new home' along the Mexican/Texan borders.
By the end of the year, the Central powers are in Bucharest, and with the failures at Verdun and the Somme in the west, and with the capture of Romania in the east, and all nations looking to another poor harvest, the Entente and Central powers sit down to discuss peace.
The net result of this is terms are drawn up on 'equal partnership', Finland and Poland become their own recognised states, Germany gets some minor border rearrangements, in exchange for releasing occupied coloniels capture during the war. Austria-Hungary remains united, with the creation of a large Bulgaria, minor Romania and Serbia.
Futhermore the peace terms preceed the Russian mutanies and fall of the Romanovs, without the Oktober Revolutions.
Foch never speaks of an armistace for 20 years.
As the 1920s begin, the Mexican Revolution is cementing itself in a popularist movement to kick the Americans out of the 'Atzlan State' that they have come to enhabit having fled the ash clouds (the American west coast is still fairly sparsely inhabited), and while not open war between America under Willson tensions between Washiton, Texaus and Mexico are large. As America enters the Great Depression at the end of the '20s, the Second Mexican-American war breaks out with Americans on the Texus border skirmishing with Mexican fighters.
The war escalates in the region, but because of the depression, the net result being in American settlers being kicked out of Mexican lands back into America.
In Europe the timely end before the Russian food shortages prevents widespread unrest against the Tsar which Lenin can capitalise on for a Bolshevek revolution on his return from exile and is eventually caught and shot by firing squad sometime in the early '20s for trying to forment rebellion, while unliked the Liberal ideas sown by political thinkers like Lenin take root over the decade forcing reforms and a consituational monarchy much like in Britain.
Without the harsh settlements of the Treaty of Versailles, Germany doesn't go into ruin, and Austria-Hungary doesn't split. However following the Russian reforms, German people call for the Kaiser to exist alongside a democratic 'body politic' sometime in the 1930s.
Similarly fascism doesn't spring up in Italy under Mussolini as Italy cannot claim and territory gains from not participating in the war. Stalin stays a Russian peasant, and Hitler never makes a show on the political screen.
With no Soviet Union under totalitarian regime, or fascist authoritarianism liberalism becomes the dominant ideology of Europe and there is never the 'great clash of ideologies' in World War II, or the Cold War. Nuclear weapons are not developed in the '40s because there is never a war or the fears of a 'Nazi bomb project'.
etc. etc. main point...all the nasty of the mid 20thC doesn't happen.
The worst?
The middle of the Cuban Missile Crisis and the people watching the Seismograph think the eruption is a Russian first strike nuke...
