I think the most important part is accessibility. As Austrem said. Being able to enter the bank in any city or town is kinda necessary to encourage social activities.
Also, a very complicated Bank system is not that necessary in my opinion, however it is important to have the tools to organize things inside, such as creating tabs, and buttons that allow ordering the different items according to rarity, type of object, etc.
A great addition could be the bank's connection to the auction house or a sales npc in the bigger cities, so you don't have to make those tireless trips to empty the bank or sell farmed items.