Well, since 1/10 of Mexico's population fled during the Mexican Revolution, almost all to the US, that's quite an issue that OTL shaped what American policy to Mexican immigration was. Even if the modern Border Patrol was original formed to keep the Chinese in Mexico trying to cross over to the US out (and failing miserably), policy makers had always clearly felt that something need to be done about the Mexican border.
But since the US is so prosperous, I doubt that even a much better Mexico than OTL (Portugal-levels at least must be doable!) won't be sending immigrants. But as you said, it'll be sending fewer. That means that the main issue in border control will be the Chinese. To this day there is still an issue with Chinese illegal immigration, and to a lesser extent, other illegal immigration from East Asia. Since the Chinese are going to be the main group illegally immigrating to the US, you'd see a continuation of the historic anti-Chinese sentiment throughout the US. Maybe not as high, since there will be less Chinese than Mexicans ever were. Also expect crime that went on/goes on in Chinatowns to attract much more attention, as crime attributed to illegal immigrants/Mexican gangs tends to attract attention. There will be a paranoia about gangs in Chinatown, and that goes double if an event like the Golden Dragon massacre occurs. Also expect the Vietnamese, Cambodians, Hmong, and other Southeast Asians in the US to fall under the eye of anti-immigration groups, or at least groups who pay huge attention to crime committed by people of those ethnicities. And when in doubt, if Islamic terrorism is still a thing (as it was against Americans even before 9/11), immigrants from Muslim countries would still be suspect, even if your average Muslim criminal in the US isn't so much a terrorist as a common street thug interchangeable with a street thug of any other ethnicity. Overall, expect anti-Chinese paranoia to be heightened, and that will extend to China itself (Communist or anything else, China will be viewed far more negatively in the US than it is today).
Also, what about Central America? A lot of the illegal immigration in the past decade is more from Central America than Mexico. Does a more prosperous Mexico instead absorb these Central Americans? Even without the brutal civil wars there, I can't see the Central Americans not emigrating to other countries, in which it would be the US first if they could, and Mexico second. Though without the civil wars there, you probably wouldn't have the MS-13 gang, whose actions have stoked a critical paranoia which many Americans have toward Central American immigrants. You'd no doubt have immigrant gangs from Central America, but perhaps none as brutal and headline-grabbing as MS-13 was when they were building their reputation.
Or more likely the Democrats would just adopt their message accordingly. Perhaps retaining more of the working-class whites in the Midwest and such who now vote Republican.