Bank of Ireland's normal banking system for personal users works fine on Mac OS X in Safari (apple's default browser) or Firefox.
I also have absolutely no problems with AIB or MBNA.
Not sure what BOI are up to with their business banking. It's possibly either old inflexible software, or they're running on some microsoft-centric system that is locked into Internet Explorer mode of thinking.
It's just poor programming in my opinion, many people thesedays opt not to use Internet Explorer (for security reasons) and given the explosive growth in Apple laptop sales in Ireland, I really don't think Mac OS X is that unusual anymore. Any time I get on the train to/from Cork I would reckon that well over 50% of the laptops in use are Macs and you'd typically see about 5 or 6 in each coach on most trains.
Also, in business environments there is increasing use of Linux, Solaris and other non-windows systems.
It just seems rather short sighted of BOI to be so restrictive.
If your Mac is one of the more recent Intel machines, you can download virtualisation software like Parallels (at a cost) and you can then install windows to run in a virtual machine on top of OS X. So, you can use IE without any issue.
NB: you do have to provide your own licenced copy of windows.
You can also download apple's Bootcamp utility and simply dual boot, but that means that you have to decide to boot into Mac OS X or Windows, you can't do both simultaniously.