I have been recording the Social Security Number in my personal database, but I have not yet added it to the profiles I've created. I have relatives with very common names, so I intend to add it in the biography to enable others to verify sources. This will only be helpful for comparing 20th century U.S. records, though.
Since the SSN appears on death certificates, employment documents, and military records (for mid to late 20th century documents), it will enable someone to verify that they are sourcing the record for the correct John Thompson or William Jackson.
But, as Anne has already mentioned, it is important that we never list a SSN for a living person.