I have tried using Firefox on my bank site. It gives me a message that the browser is not supported.
Is javascript turned on? You might get that message if it isn't, and most on-line banks require it.
Now surfing with JS turned off is smart, and it is a pain to have to go back and enable it when needed and then turn it off again when you're done. That's why the Firefox extension "NoScript" is so great. You can globally deny JS, and then selectively allow it for certain sites (every time you visit), such as on-line banking w/out having to go through all the hassle.