SSN's are a critically important identifier, not only used for social security benefits, but to secure credit information, medical records, etc.
There has been efforts to protect SSN's by hiding them, but this has done little to thwart fraud, while making them far less useful as a personal identifier, and creating risk for service providers who store them. The recent Anthem data breach is a good example of the problem.
Instead of hiding them, why not secure them. Perhaps separate them into two parts - the currently assigned number, which is public, and a new number which is private. Much like having a PIN on a debit card.
PINs would never be stored, and would be required for any transaction, such that others with the SSN would not be able to use it without the corresponding PIN.