They were designed to be only used for the administration of social security. Since they were sending monthly checks, they needed a way to know that the person going to the office and saying their address changed was who they said they were. This was at a time before driver’s licences were common and they didn’t have any other type of ID, and there were just a lot fewer people.

Later on the SSN started to be used by banks and other entities even though it was never meant for that, and the risks associated with the relatively insecure design just compounded, because instead of just fraudulently claiming someone else’s social security checks (which, unless the target died, would probably be figured out within a month), it opened up all sorts of extra avenues for fraud.