First of all they are already required by federal law to check documents. It is already a $10,000 fine per illegal employed. But the objections of difficulty checking documents is very well addressed in H.R. 4437. There are very reasonable protections for employers in cases where the gov. verifies a worker's documents and they are later found to be fraudulent. The database is already in place too.
I hate the arguement regarding checking a persons legal status. With the technology we have today it would be very easy. In fact it can be done at the private sector level. If you bounce a check to the grocery store it is known thoughtout the country within a day. Certainly if we wanted to get serious about this we could.