Besides, it's a really bad form of ID. The numbers aren't even unique, and up until 2011, a few digits are reserved for geographical information.
https://en.wikipedia.org/wiki/Social_Security_number#Structure
If I had to reach for a hasty solution, it would be to use the IRS taxpayer ID instead. Of course, that might weaken the SSA's importance overall, so that wouldn't be without consequences: https://en.wikipedia.org/wiki/Individual_Taxpayer_Identification_Number