Employers can verify social security numbers electronically
Did you know that employers can verify that a social security number belongs to the person who's applying for a job and not to someone else, and that they can do it electronically? The program is called "Basic Pilot Employment Verification Program", and it's been available in California, New York, Florida, Illinois, and Texas since 1997. Nebraska was added in 1999, and all the other states were added in December.
Given those facts, you might say, "why, that could make it much more difficult for companies to plead ignorance in the unlikely event they're found to have hired an illegal alien."
Ah, but there you'd be wrong: the program is completely voluntary. Not only that, it's received little publicity and it's used by very few employers. Apparently, out of 5.7+ million companies in the U.S., just 4,385 use it. In Arizona, it's 101 users out of 96,000 companies.
More details in "Few firms use migrant ID service". As for the competitive disadvantage suffered by patriotic companies that use this service and don't hire illegal aliens, well, that's why we have courts.