It doesn't even have to be strictly a border patrol issue, the illegals can be stopped by fining the businesses that employ them and making it unprofitible for them, by tracking down the illegals after they enter and deporting them - which would send a strong message, and by making it impossible for illegals to get drivers licenses, educational benefits and welfare programs. But it all has to begin with somebody at the top giving a hoot. Immigration is good, unchekced illegal immigration is self destructive.
I've heard that suggested before, and that particular idea doesn't sit well with me. It should not be the job of a private employer to investigate its employees to determine citizenship status. That's government's job, not the private sector's. And immigration laws should only inconvenience immigrants, not citizens.
A better approach, IMO, would be to keep the EEOC off the backs of employers, so that employers don't feel the need to beef up on their "diversity" ratings by hiring illegals.