You need to specificially enable Java in Safari preferences (Safari -> Preferences -> Security -> Enable Java). Firefox may have a similar option.
You can test whether Java is installed and working by visiting: http://javatester.org/version.html
Java IS enabled both in Safari and Firefox - and the bank applet still does not work despite re-installations and reloading of browsers.
As I am on 10.7 and must have started out with Java 6. As per below I must be a 'big loser'... Any answers to this?
On this link (http://blogs.computerworld.com/application-security/21173/ugly-side-latest-java- updates) I found this information:
Macs running Lion (10.7) and Mountain Lion (10.8) are in a more complicated situation.
These systems shipped without Java and users could have either installed Java 6 from Apple or Java 7 from Oracle.
A system that started out with Java 7 from Oracle should be fine. Install the latest Java update and you're done. It is, however, unlikely that many Lion and Mountain Lion systems started out their Java life at version 7. Up till now, Java 6 has been provided, by default, by Apple.
Systems that started out with Java 6 are the big losers in the Mac world.
Apple's latest update to Java 6 takes the drastic step of disabling Java in all browsers. Java 6 is thus only available to installed applications. Hopefully, that's sufficient for you.
If, however, you need to run a Java applet on a website, then Apple forces you to install Java 7 from Oracle for just that purpose. This results in both Java 6 and 7 being installed and, no doubt, permanent confusion as to what software is using which version of Java. Not to mention being on the hook, going forward, for updating both versions. For more on this, see Apple gets aggressive - latest OS X Java security update rips out browser support.