Hm really? I can't imagine a bank in 2014 getting away with this, at least for consumer-facing services. They are just going to write off all their mobile customers?
If you're talking about B2B or other more back-end type stuff, yeah some of this is annoyingly outdated. My bank's "business bill pay" website looks like it was written in 1998 and not changed since.... it does work in FF and Chrome though, since it's just bare-bones html forms and almost no javascript.
No, consumer facing websites. I work closely with a Service Desk department. I've dealt with having to get major bank websites up and going one too many times!