Avoid US Entry into WWI could well do it. It marginalized many of the groups most opposed to prohibition so that they could not effectively stop it and it effectively put the country on prohibition early thanks to War Prohibition
That or have the intent of the Volstead Act spelled out beforehand. A lot of pro-Prohibition people thought it went way too far and were surprised at hell at the banning of everything over .5% ABV, War Prohibition had only gone to 2% ABV, and that it included beer and wine rather than just spirits. If the public realizes the truth it likely won't pass
One could also avoid the 16th Amendment, as without that the response is "we can't afford that" as without an income tax something like 30-40% of US Government revenue came from taxing liquor
Despite relying heavily upon the "USA does not enter the war" POD, I have mostly ignored such big butterflies to the USA. First, Wilson not running and another Democrat such as Clark being President, next the failure to pass Income Tax and/or some of the early legislation such as the Federal Reserve, and lastly the war no war impacts the economy, culture and passage of Prohibition. I think to keep the USA out of the war, no Wilson is a more certain bet, but it unwinds some big potential butterflies. Again I have simply ignored them. Moving through a neutral USA, the potential is to me a toss up whether butterflies overtake Prohibition or if it is independent enough of a thing that it prevails in spite of these butterflies.
Temperance was a murky social engineering moralistic drive that seems all too American, despite the values of limited governance and personal freedom, lurking in the shadows is this Utopian urge to make men "good' through any coercion necessary. My sense is that without the pressures of war, Prohibition has only a slightly less chance of passing, for me the ball was rolling, but oddly I think that Suffrage might have faltered. And that is a bitter irony. Without the anti-German hysteria but existing antipathy to a distinct minority, I would think enforcement gets even less enthusiasm in many ways. Prohibition might be an even more scoffed at law, more openly defied or ignored, with the Federal authorities having not gained the same wartime expansion in power, the thing needs State power to work, and the resulting struggle between Federal and State and Local police powers could be messy. If we let Prohibition fail, the butterflies are many and interesting.
First, it might quash Women's rights and voting in many ways, it might derail the anti-German and by extension anti immigrant tenor but shift energy to private hate through the KKK, it will forestall the rise of a big Federal government and police power response, it changes the Roaring 20s, kills the Gangster legend, quashes the noble criminal motif, preserves industry and revenue that moderate the Depression, and on and on. I cannot tell what thread pulled unravels Prohibition or once undone how that reshapes the USA. Every time I think about it the image of Mothra appears and America is buffeted mightily.