Norwegian BankID (used by most banks in Norway) uses a java applet for login.
The current combination of java, mac osx 10.8.x, applets and BankID is a mess.
Things to note:
Java 1.6 from Apple removed support for web applets - so you'll need Oracle java 7
Java 7 from Oracle will work - but - not in Chrome (Chrome doesn't support 64 bit java - Oracle don't provide 32 bit)
Most BankID pages test to see if java is enabled - and after installing java - it still gives the java not installed or enabled error. The issue seems to be that you have to activate the plugin by loading an applet that doesn't test for support - for example the one on http://www.java.com/en/download/testjava.jsp prior to trying to log in to the bank
BankID state that there is an issue in Firefox where the OK button does not get enabled - you can get around this by holding the CMD key in for a few seconds until it enables. This is claimed to be due to a bug between Firefox and java. Note that in my testing this is exactly the same issue in Safari and the same workaround works.
So - install Oracle java, go visit a test page - log in with a workaround to get OK buttons to enable and you might just be allowed to login to your bank.
The BankID applet needs to die.
Sources: