First off I think you would have to place the starting point for this TL after the 1929 crash and during the early years of the Depression. Maybe Hoover's inept policies would have sparked a strong socialist party. Also, FDR must be out of the picture. A Hoover-like successor, with similarly inept skills, would provide greater impetus. I don't think this phenomenon would last past WWII, as the wartime economy ended the Depression and highly stimulated the American economy.
As mentioned, this socialism would have to be a market-based socialism, sort of a capitalist-socialist hybrid like many European nations. I'm surprised that a socialist movement did not spring up during the Depression, but I agree that the USSR had a lot to do with that.
Imagine McCarthy's response to the Red Scare AND a viable American socialist party? Freaky ... that'd be an awesome scenario!