those illegal immigrants will keep the wages low because they will work for a lot less The people who keep the wages low are the people who pay low wages. And in the case of people paying wages that are lower than the statutory minimum, the authorities who fail to catch them.

Of course they couldn't do that if there weren't workers desperate enough to accept the jobs for that kind of money, and in the case of illegal immigrants subject to blackmail. But "he who pays the piper calls the tune" - and in this case that means the employers who pay these wages.