Not all that hard, honestly. The Socialists were a growing party up until the Wilson administration used World War I as a pretext to destroy them. And even still, they reorganized and grew pretty big again during the 1930s before Roosevelt stole their thunder. So there are two possibilities here, I think.
No. 1 - The Socialists build up during the 'teens, take power during the thirties
Assuming that WWI still happens but the U.S. doesn't get involved, the Socialists will probably start getting people into Congress around 1916-18 and get a hell of a lot more into office during the 1920s. Their antiwar position helps a lot here, and Democrats upset with Wilson combined with Republicans upset with the rightward shift in their party will be the party's main benefactors. Assuming that WWI ends around the time it did IOTL, with an Entente victory and whatnot and we still get the Crash in '29, the Socialists probably take the House in 1930 (I can see them becoming the second largest party sometime in the 1920s in that chamber, honestly) but don't take the Senate. Then you probably get a Socialist President in the 1930s. I don't see the Socialists getting a majority in the Senate until the 1950s or 1960s, though. The Democrats will become a de facto Southern interest party, and will become a third party before most of its members fold into the GOP.
No. 2 - Socialists beaten down, but rise during the Depression
History goes as it did IOTL until Zangara is able to kill Roosevelt. Garner's conservatism allows for a more radical alternative to develop to the existing order, but the GOP is the beneficiary in the meantime. Socialists, Farmer-Laborites, Progressives, etc. form some sort of alliance near the end of the 1930s, but don't become a major party until much later. They probably wouldn't be called 'Socialists', either, but I see them as being enthusiastic members of the Socialist International. Maybe they'd be 'Progressives'? Not sure. But I feel like the Democrats still end up falling into the GOP and the main two parties are a more liberal GOP and the Socialists. Maybe the Socialists win the White House in the 1960s?