One side effect of all the Java updates that have been issued lately has been that some poorly-maintained Java applets have stopped working. The only solution, if this is the problem, is to request that the applet be updated.
Ask your bank's tech support if they support Java 1.6.0_33. If they don't know what you're talking about, ask to talk to a more senior representative. Continue this until you get someone who knows what you're talking about and can help you. If no knows what you're talking about, switch banks, because you don't want these people in charge of the electronic security that protects your money!
Same thing happened to me with Safari and KeepVid.
I got a "missing plugin" message.
Try downloading directly from Java:
Install and then restart Safari.
Fixed it for me.