The bank had armed guards at the entrance who stopped anyone carrying weapons from going in (except of course our Dwarf, who claimed his double-bladed axe was merely ceremonial). Inside the bank, there was another armed guard and a magician who was now and then casting 'Detect Magic' to see who was using magic. Then the client would be recognised by face by a clerk who would then count out the money.
There are spells that make someone look like someone else -- so magic easily would defeat the identification process. Of course, you'd have to defeat the Detect Magic -- but it is also possible to have an Alchemist put a spell in an item, creating a magic item with that particular effect. Sure, the item is detected with Detect Magic, but your average adventurer or magician would light up like a christmas tree when viewed in such a way. I think my character, who is not especially keen on magic, has some three or four magic items on him. Other characters have even more. Some characters have magic jackets -- would those not hide anything magical under them?
So the fact that someone is carrying a magical item is not enough evidence. The magician in question also has to be able to determine what spell is being used in the magical item -- which is a separate feat. And he has to do this constantly. Sure, high-level magicians can cast these things at will and have a good chance to identify the spells in effect. But if you have to have an army of specialist working around the clock, it just doesn't scale anymore. A magician that spends all his daily magic at the bank has to be handsomely compensated -- he could make loads of cash by selling his magic to the highest bidder or by adventuring. Surely the bank should match that price.
'Detect Magic' (page 219 of my 3.5 PHB) lasts 1 minute per level, but the cast has to concentrate. So a level 10 mage can detect magic for 10 minutes per spell, if his concentration is not broken (by a loud noise? Someone talking to him?). A level 10 mage can cast 20 spells per day. So that makes for 200 minutes (slightly over three hours) of security. Suppose the bank is open 8 hours per day, that would mean the bank has to retain three level 10 mages!
My character is level 8, and has, next to a wide array of various expensive weapons, trinkets and artifacts, about 2000 gold pieces in cash. A mage who's 2 levels higher should make much more. So in order to be secure, the bank has to charge enormous interest on loans, or something. It can't be commercially viable!