Ok, I'll be honest. I'm pretty fine with how we handle money now. Sure, back in Everquest, I needed to go deposit my money in my bank, because due to the encumbrance system, if I had kept all my platinum coins on me, I would have been glued to the spot, unable to walk. But in WoW, there's no encumbrance, and our coins take up no inventory space.
Still, I've seen a lot of people ask to be able to deposit money in the bank, so, being an equal time type of guy, I thought I'd sit back and try to figure out what the advantages would be.
As far as I can tell, there's basically two. One, it prevents you from making impulse purchases at vendors or in the Auction House, and two, if the money bank is shared between all characters on a server, it makes transferring money between characters for major purchases a little bit easier.
The first, well. I suppose the whole impulse purchasing thing can certainly be a problem, especially if you're leveling up a new alt. Sure, that level 29 "of the Monkey" cloak is up for 10 gold, but damn if it wouldn't be awesome for your Rogue. Sure, that Orb of Deception is up for 750 gold, but it lets you turn into a Gnome. How sweet is that? In that case, being able to "lock away" part of your gold and making you make a conscious decision to walk to the bank and withdraw it before you could use it could prevent those. If you don't have the gold, you don't make the purchase. If you have to go get the gold, you have an opportunity to think about it and decide if it's really worth it.
The second is also understandable enough. I recently purchased epic flight for my Hunter, and do to so I ended up logging between 3 other characters to send over enough gold so my Hunter could afford it. So in that case, being able to walk over to the bank and just grab gold my characters deposited back the last time I was actively playing them would be convenient.
That said, I'm not sure either of this reasons is really good enough to put Blizzard on programming the option. The fact is, you can already more or less simulate the idea of a bank with a bank alt. Send all your gold to them, leaving yourself only a bit to buy food, water, ammunition, reagents, and any other emergency supplies. The process of switching over to them to withdraw gold to buy something should give you the same "cool down" period for impulse purchases, and since you know all your gold is with them, you will always know exactly where to go to send your latest 70 the money to buy flying skill. I suppose it can be a pain to switch to your bank alt to get the gold, but hey, it leaves the programmers free to work on WoTLK.