Banks are used to store items that are purchased with Aeria Points via the website or through a promo. In the case of participating in a promotion, the buy x amount, get x amount free of the same or another item, the free item(s) will be transferred to the bank teller and be available to be picked up at any time.
If you are looking to transfer money from one character to another, you have to ask a friend to help you with the process, done through a character to charachter trade* and x amount of gold given to a friend. Then you get on your different charachter and ask the friend to trade* back the money.
*Nothing has to be traded in order to give another player gold for a transer/payment/reward.