Republicans can hardly claim the high ground on this. Republicans don't care about the "working class" either - they say they do, but only in the context of immigration and trade. In other words, when foreigners are involved - red meat for America's xenophobic 100% white small towns. But in other areas such as healthcare, benefits, and education, as far as they're concerned, the working class can go to hell.
Remember that a lot of business Republicans do support immigration, of both sorts (though they don't always admit it). But unlike with liberals and progressives, it's not for altruistic reasons; it's sheer economics. Many firms are willing to turn a blind eye to letting illegal immigrants come in, so long as they
stay illegal, for these firms like having a permanent underclass for the cheap labor. They look at Saudi Arabia and the UAE and see them as models to emulate.
On a separate note, with the "Democrats support increasing immigration because they want to import Democratic voters" arguments xenophobes spew, you do realize that it can be flipped to "Republicans support restricting immigration to maintain Republican vote share," right? And with the authoritarian practices that the GOP is doing right now in America, the latter reasoning seems much more prevalant than the former.
