The Banks building

works similar to the general:cost upgrade. It reduces the upkeep of all units in the same city as the bank by 1. That's basically it, if you have for example, 50 units in the same city as a bank, it will reduce the upkeep of those units by 1 each, so you save 50 cash that turn.
FAQ
Does the Bank building make units cheaper to buy?
No. The units all cost the same to buy with and without the bank building.
Does the Bank building have an upkeep?
Yes. If you are None, the bank costs you 69 a turn in upkeep.
Does the Bank building work on units built from other cities?
Yes. The upkeep reduction is on ANY unit in the city with the bank.
Does the upkeep reduction stack with the general's upkeep reduction?
Yes
Does the upkeep of buildings get reduced?
Yes, even the upkeep of the bank itself gets reduced by 1. Note the bank's upkeep is 69 (70 - 1) if you are none strategy.
What is the break even point where the bank is worth buying?
If None is your strategy. You need at least 70 units for the upkeep reduction of the bank to outweigh the upkeep of the bank itself. If you have 70 units, in a city with a bank, it takes 600 turns to pay off the investment of the bank.
If you have 79 units, it takes 60 turns. If you have 169 units, it takes 6 turns to get a return on investment.
When you are playing games, if you have large stacks with hundreds of units and you tend to keep them in one particular city (like your capital) often, then a Bank might be worth buying. A useless tidbit about banks and general cost reduction is that they reduce upkeep by reducing the "cost" stats of all units in the same stack as them. Generals with the upgrade reduces the cost stats by 50 and the Bank by 10.