Maybe if you measure it by what hit radio songs they remember from childhood, but the things most people would consider more fundamental paint a different picture. If you take this source I haven't vetted[0] (I'm getting this from common knowledge, not surveys, to be honest) the evidence seems pretty clear that immigration from south of the border is actually supporting our most longstanding conservative cultural institution, far from "diluting" or "eroding" it.

If any reader is curious, the irony only developed around the time a certain presidential campaign normalized anti-immigrant rhetoric; until then the "old conservatives" were fully aware of the importance of immigration to business owners and the lack of any real cultural incompatibility and never took any steps to reduce it. They sided very much on the elimination of the welfare state which was the solution they saw for the supposed cost issue (highly questionable to begin with, in a country with as little in the way of social programs as the US after the 1980s drew to a close. In a system designed for people that thought social security should depend on the number of hours you have worked, it is hard to imagine what costs they'd be objecting to.)

The way the shift in alignment was carried out always struck me as a sobering example of the shortness of memories, or the impermanence of the public's convictions.

[0] https://www.prb.org/resources/immigration-gives-catholicism-...