One of my first jobs was a bank teller (no degree needed). Also, there are Personal Bankers who also do not require a degree to do their job. However, like many employers, if you would like to get promoted further into management, a degree is usually required.
In the US a degree is not normally required to be employed in a bank. However for more prestigious and management positions a degree in business and bank, especially international banking, would be a major asset.