So you're talking about prosecuting 3-4 million employers. Good luck.
No, I don't think you really need to prosecute 3-4 million employers.
Employers aren't stupid and they aren't incapable of change. I think that all you need to do is to increase the cost of (penalty for) hiring illegal immigrants to a level that rational employers will decide that it is in their best interests to quit hiring illegal aliens.
We're talking about rational businessmen here, not addicts. They care about costs - that's why they're violating the laws now.