The main point of a banking system in a post-apocalyptic setting would be to ensure the safety of your cash - instead of travelling loaded with caps and risking getting mugged on the way, you leave your cash in one place
It used to be that you would pay interest to the bank to have them safeguard your money as well.A better idea might be to pay a fee to the bank to issue players "traveler's checks"--you can cash them in at any merchant or base construction guy, but if another player kills you and loots you, they won't be able to cash them!
I really doubt ncr will go belly up