What if... We had not shut the door on the immigration wave in 1921?

[Skill and Chance]

Assuming the door remains as "open" as it was in the 1910's straight through to the present day, Dem dominance is even more dramatic during 1932-68 than it actually was.  It's entirely possible that the GOP never recovers from the 1930's and gets explicitly replaced by something like Europe's Christian Democrats by 1945, but either way, the right's platform would move much faster and more decisively in that direction.  In 1968, a conservative running on Nixon's platform would have been much stronger, probably 2 landslide wins in a row and they flip congress at some point during that.

I am really not sure how it would impact the elections between 1976-2000 if they occurred between the same 2 candidates.  More likely, the Reagan-Bush and Clinton-Gore political movements just never happen because their coalitions simply wouldn't be large enough to compete.  The fiscal conservative push may simply never happen and we just go straight from Nixonism to Trumpism, with wealthy suburbs and tech workers ending up in the Dem coalition much sooner and Republicans holding onto Nixon's gains in the industrial North and making it their base. 

Society would be less socially liberal in general.  I doubt Roe v. Wade and Obergefell would have happened, but some states would have legalized abortion and gay marriage on their own.