Had Mexico gone communist before WW II, it would have been tolerated given the general attitude in the US at the time toward left-wing politics. Afterward-and especially after Churchill's Iron Curtain speech-I could see where a very real physical barrier would have been put in place. And I mean the entire length of the border. The border would be very tightly controlled and you better be able to show some damn good reasons for crossing. Cross-border trade would probably diminish to a trickle, and immigration would do likewise (although illegal immigrants, somewhat ironically, would probably be viewed as refugees from communism and would be welcomed, assuming they integrated themselves into US society, starting with learning English and securing employment).
Once the Soviet Union collapsed, maybe the border would open up some but it would take someone along the lines of Reagan or Nixon to make that happen. Even today, I suspect it would still be tightly controlled with a bare handful of checkpoints: say, maybe at the larger cities (Brownsville; Laredo; El Paso; San Diego) and a handful of interim points in California, Arizona, and New Mexico. The border towns of Calexico and Mexicali would probably glare at each other across the wall and have nothing more than that to do with each other.
Given that refugees would be a relatively small cohort, one wonders where inexpensive labor in the US would come from for jobs that are commonly viewed to be held by Mexicans? Thoughts?