Binos, does it tell you which plug ins are needed? Sometimes they need to be loaded manually instead of through the automated installer (although it should then tell you so!).
You could have a look on the
Mozilla.org site. A lot of plugins are available there.
I always get pi$$ed off at sites that refuse to work and actually tell you that you need to use IE!
You could try getting in touch with the helpdesk for the internet banking to see if they have a solution, but I wouldn't count on it!