The leaders of the right don't want to reduce illegal immigration for two obvious reasons. First, the country is better off with those people here -- we have a dangerous demographic age imbalance, and the disproportionately young workers who come here illegally help to mitigate that problem, so we'd be idiots to try to cut that off. Second, the more immigrants who come here, the further to the right the country will go, politically, since racists and xenophobes get triggered by seeing more brown people, and that makes them easier for plutocrats to manipulate into voting for pro-plutocratic policies. There's a reason that the most liberal countries in the world are fairly ethnically homogeneous (like Scandinavia), and that the US moved to the left during the era when we had the least immigration (the 1930s through 1960s).
So, the game of the GOP is to complain constantly about immigration, to keep the xenophobes stirred up, while being very careful not to do much about it. It's helpful to them to have a few showplace concentration camps, and the like, to give the racists the thrill of seeing cruelty against "those people," but they wouldn't want to do anything that might have a big impact (like, say, throwing the book at the illegal employers who lure people here in the first place).
