Woah, there standard banking must be different from business then!
I use barclays for business banking and barclaycard business (credit cards) and I do not have Windows (So no ability to use IE) and have logged in from Firefox and Seamonkey in the past with no problems.
Generally to login and do banking I can use any browser (I've never been stopped yet). They simply ask for Surname, Membership Number, Last 4 Digits of card, and tell you to use indetify on the PINSentry card machine and give them the code it gives you.
From there on you can do pretty much anything such as transfer money etc.
Never had any form of activex prompt or request to use internet explorer tbh.