Check with your bank if they offer internet banking for mobile phones.
My bank did use java for authention to get to the internetbank services, so I could not use it on MorphOS either.
I did contact the bank, but they said that it was their only option. No java - no banking.
But then I subscribed to their cellphone banking services and they sent me an SMS with an URL. Then I retyped the URL in the OWB browser, and then I could log in and use a simplyfied version it - without java. The page looks very simple, and I guess it might work even with IBrowse.
btw, isnt Netsurf available for 68k as well? Not sure about the requirements though.
