Banking system for items could be like this:
For a certain amount of trias per week/month/day a player gets a small room, which has a stong lock, nearly unpickable, and into that room he can drop the items: on shelves, boxes etc.
Money deposit should be like usual, talk to an NPC, give him/her the gold and he/she will add it to your account.
Another idea is making the banking system like in real life, that bank can use the deposited money to give out loans, trading to increase the money,make things better in the bank etc.
Also there could be player run banks and smart peaople can get rich by running a successful bank.