I'll just chime in and say that this is not necessarily true. At least in my area.

There is no state or federal law that says I can't conceal-carry inside a bank. It's up to the property owner to make these rules. Because it's relatively difficult to get a carry license in New York, most businesses don't bother posting a no-gun policy at the entrance around here. This includes banks. In the south, it seems more common to have these policies posted even in restaurants.

I keep a TD Bank account open just for cash deposits. I can then pull from TD to my credit union. The TD branch I use most often has been held-up at the drive-thru teller window twice this year. A no-hats policy wouldn't have helped that and they still don't post a no-guns policy at the entrance. There is no security guard either which is something I've never seen at any bank I've been to in this area.