With a start somewhere in early 20th century let a socialdemocratic movement grow strong, and have it take, more or less, the role of the OTL democratic party. AFAIK unions were fairly strong in USA in early 20th century, but radical elements of all kinds remained relatiovely strong and thereby legitemized anti-union forces to basically crush the unions as a serious political force. Let the socialdemocrats win the fight over the unions early on (like in most European countries) and the big popular centre-left party would likely be socialdemocratic, and still keep the main OTL developments.
FDR's New Deal could no trouble be a socialdemocratic project as could his line up to and during WWII. I guess the biggest difference will be trade unions playing a bigger but also more constructive role in US society. McCarthy will not have many chances, as the socialdemocrats will prove much more effective commiefighters than he and his types ever could hope for.
The governments of Kennedy and Johnson could also, added with some mainly ceremonial stuff like celebrating 1st of may, be quite close to socialdemocratic. I guess USA will by mid 20th century have a comprehensive public health care system though.
(writing so while officially celebrating a day off, but like so many others mainly working at the PC...not very loyal to the labour movement's founding fathers
).
Regards
Steffen Redbeard