Check the Ts & Cs
I think that if you look at the terms and conditions of most online banking services, you will find that they have a list of known and supported OS/Browser combinations, and I would be surprised if any Linux platform is listed. This gives them an immediate get-out from most Linux users.
My primary bank would like me to install agent software on my machine (at least last time I looked) to access their online banking system. Of course, this is windows based.
And the AC who was talking about Linux viruses has obviously not taken into account how short the wikipedia page about Linux viruses actually is, nor has he looked at the viruses listed. Many of them are old definitions, some are for products not involved with browsing, and virtually none of them will cross the user/system boundry unless you are stupid enough to be running the vector as a privileged user (root).
Of course, Linux is just as vulnerable to social engineering (i.e. Phishing) attacks, but that is because the user is being targetted, not the OS or browser. In theory, it is possible to install anti-phishing plugins in Firefox, but such defenses are only as good as the block database that is being referenced.
I'm just waiting for the banks to insist on content filters being mandatory for their services. When that happens, the simple port filter firewalls implemented by most routers (and Linux Tables and Chains firewalls) will not satisy their requirements, and we will be further beholden to Microsoft.