"Why would they be paid on the books"
I envision a system where a company could 'sponsor' a guest worker to come and do work...and harsh punishments for companies that pay immigrants under the table instead. There is something called e-verify or e-check (don't remember exact name), which is making it harder for companies who do contract work for governments to get by. I know someone who owns a concrete company...and he had to do some record checking before getting a contract...and the status of one of his employees was challenged (ultimately proved he was legal). Anyway, there are ways to make it unbearable to pay immigrants under the table - monetary penalties, black lists for public contracts, etc., and these methods should be used.
BTW, on an unrelated subject, I recently reported to the IRS a contractor who was paying his employees as independent contractors (not withholding his half of the social security). I just got a copy of the letter he was sent...a quick summary would be: You are fucked
. So, if people cheating the system are reported, it seems action is taken.