I understand your concerns. However, usually landlords require this before you move in or are even given a lease - the SS# is used both to confirm identity (although it is not technically a form of ID) and to run a credit check. As such, don't be surprised if the failure to provide ends up meaning you will not have your lease renewed. Because in many rental situations you'd have had to put your SS# on your application, you may not be gaining much by refusing this time.
A landlord can rent on his own terms and require different application procedures. If you feel uncomfortable doing this, say so and if he insists, it may be time to say goodbye!