Cross check all Social Security numbers for IRS contribution activity against those in deceased or incarcerated database.
This will show discrepancies in activity for numbers that should be idle.
These could be blacklisted and researched for fraud, as the vast majority are sold via black markets targeted at illegal immigrants.
A SS verification period along with I-9 verification before hiring associated with stiff penalties for offending businesses would discourage the current practice of hiring cheap labor without penalty for compliance.
Cleaning up and securing the Social Security system will be necessary for any progress in this realm.