Many are ignoring the problem of forged documents. I was an employer in California until 12 years ago. I was virtually certain I had illegals on my payroll, but they had "green cards".
Perhaps things have changed since then. (Is there now a system for an employer to check for forged credentials?). But I suspect it's gotten harder, not easier.
You're right. Thsy get their fake papers on the way up here. It wouldn't be practical: the only way to confidently avoid hiring illegals would be to avoid hiring hispanics, and that will get employers sued for racism.
Unless the IRS tells you that their SS number is wrong there is no cheap or easy way to know.
You can run background checks on all your current employees. At about $200.00 a pop that can get pricey. Even then there is no guarantee as the paperwork may be valid but not theirs.
It is one thing if you are flouting the law and another if you have take reasonable steps and get snookered.