Paul Krugman sums up the Republican Party's schizophrenic relationship with immigration:
Republicans, on the other hand, either love immigration or hate it.
The business-friendly wing of the party likes inexpensive workers (and
would really enjoy a huge guest-worker program that would both provide
such workers and ensure that they can neither vote nor, in practice,
unionize). But the cultural/nativist/tribal conservatives hate having
these alien-looking, alien-sounding people on American soil.
So immigration is an issue that divides Republicans one from another,
not within each individual’s heart.
George W. Bush was the classic pro-immigration Republican. Whether he should be applauded for this is highly subjective. On the one hand he should probably be given credit for refusing to demonize immigrants for America's problems (a classic argument of the Right) but similarly, his motives were purely driven by profit for his corporate chums and campaign donors.
My views on immigration are complex. Importing cheap labor usually doesn't work out well for people at the lower end of the economy, so in that respect I can understand why it is a problem for many people. Wages are driven down and working hours go up. People end up working harder for less, or in many cases, not at all. The solution, in my view, isn't to stop immigration though. The minimum wage should be far higher than it is and labor laws should be enforced very strictly to make sure that no one hires undocumented workers. Greedy corporations ensure there is a constant demand for cheap labor, and if they were forced to follow the law the demand would dwindle significantly.
Also, it would help if America didn't force neighboring countries (ie. Mexico) to buy American goods through trade agreements like NAFTA that drive local producers out of business. Mexicans are streaming across the borders not because they want to be citizens, but because there are jobs to be had and none at home.