Not if the penalties for hiring an illegal are prison time. Those jobs would dry up overnight if a few plant managers and CEOs being handcuffed and walked to the back of a police car where shown on the news.
What if the illegal immigrants are using Social Security numbers that they have stolen from dead and elderly people? It will be kind of difficult to bust them.
Like I said, if we become more stringent, then there will be a market for fraud in order to "legalize" the status of illegals.