To find employment cleaning banks, visit local banks, and ask if they hire their own cleaners. If they do, ask to apply for a position. If the bank uses a contracted company, ask for the name of the company, and apply for a job with that company. Talk to all the major cleaning companies in your area that work with banks, and follow up until you get an interview and a job offer.
If you have no previous cleaning experience, you may need to work for smaller cleaning companies and not be specific about the locations you clean in order to get experience. Have a valid driver's license and a reliable mode of transportation. Banks are highly coveted contracts, and you will likely need to pass a background check before you can get to work in one. If you have any felonies, recent drunk driving arrests or anything that raises red flags, banking executives will not want you working in their facilities.
Some cleaning companies belong to a union, and they require you to join the union before you start working at the bank. This depends on the specific company and the state laws where you live. Find out during your interview if union membership is required to work as a janitor in the bank you are interested in, so you know both your entry requirements and the requirements to keep your position once you have it.