Why do banks prohibit storing cash in safe deposit boxes, and how would they know?

It is not that they would never know, they do this for a couple of reasons, the primary one is CYA - Cover Your Ass (i.e. their ass), meaning, if cash is found in your locker or if a locker was vandalized or stolen from and you claimed you had US$ 200,000 in cash, your claim would not be upheld as cash was not allowed in the first place.

Secondly, they want you to know and want to deter you from placing cash, as these would be proceeds from black money, undeclared, or unlaundered money. Banks don’t want anything to do with that.