In addition to the Italian bankers in the 14th century, banking also arose from the Knights Templar as far back as the 12th century.
The Knights Templar, being founded to protect holy sites in Outremer (the Christian kingdom of Jerusalem) and travellers going between Outremer and Europe, they had a major network of fortifications linking the two territories, and had accumulated vast wealth. Wealthy travellers who did not want to risk carrying a fortune on the road, nor want the encumberance of carrying wealth could visit a Templar stronghold and deposit their wealth, and would recieve a letter of credit in return, which would be honored at any other Templar stronghold for withdrawal.
The downside of this is that they became so wealthy that the cash-strapped Phillip IV of France wanted to seize the assets of Templars on French soil, so on October 13, 1307 (the origin of the superstition about Friday 13th being unlucky) he had all Templars in France arrested, and under torture (routine interrogation for the time) they confessed to various heresies and atrocities, and the Templars were disbanded and their assets forfieted. Phillip IV also levied a 50% income tax on clergy and was fond of arresting Jews and declaring all their assets forfeit, so the Templars were not his only targets.
In an interesting little footnote, the last Grand Master of the Knights Templar was burned at the stake, but before he died he laid a curse on both Pope Clement V and King Phillip IV to die within a year, and both were dead within only a few months.
They sidestepped the usury issues by charging "rent" on space in their fortresses. Want to borrow some money? Well, we'll loan you the money for free (since we can't charge for that), but also charge you rent on the space it took up in the keep since you're using the money that belongs there.
Essentially, for banking to exist you have to have an organization that is wealthy enough to lend money and fill withdrawal requests, respected enough for people to trust their fortunes to them, and powerful enough to protect their vaults against theft. With all the fantasy anachronisms in D&D, a large religious order establishing a crude banking system with letters of credit, possibly divinations to verify important transactions, and a chain of fortress/temples acting as "branches" is hardly a big step. I mean, the Romans had a relatively advanced system of plumbing and we don't see many aqueducts and toilets in the typical D&D world.