The key to identity theft is letting every Tom, Dick and Harry demand a social security number for any purpose under the sun. The practice should be illegal and the original wording of the cards invoked: "Not for Identification, For Social Security Account Identification Only."
Then every American who feels their number has been compromised should be able to request and receive a replacement number.
I expect counterfitting is an even bigger problem. And I recall reading that Iran possesses one of the same model printing presses which the Treasury Dept uses.