Immigration these days is mostly about cheap labor. It's great for employers and bad for everyone else. Do we really think adding large numbers of low wage workers will help anyone but employers. And do we really think it's any accident H-1B, J-1, etc workers come overwhelmingly from low wage countries (forget Japan and Western Europe)?
And btw, recent immigrants rarely vote Republican. Even Sen Amnesty McCain couldn't change that. Immigration restriction policies would help the entire country except employers. And it would get far more votes than it would lose.