Throw the bums who hire illegal aliens into prison. Problem solved overnight. The same thing applies towards the H-1b visa. Although the laws need to be strengthened, enforcement would leave us with a program doing what it was originally designed to do.
The spoilers in all of this are corporations. They break the law, yet when ICE rounds up the illegals I don't see very many people in suits being hauled away.
When will be watching the first CEO on Court TV being tried for illegal employment practices? Not any time soon, I predict.