This may be affected by how Safari is loaded. By default it loads as a 64-bit application. Quit Safari. Select the Safari application in your Applications folder. Press COMMAND-I to open the Get Info window. You should see a checkbox labeled, "Open in 32-bit mode." Click the box to check it enabled. Close the Get Info window and relaunch Safari. Now try your bank site to see if this fixes the problem for you.
I didn't see this as being run under Safari, but Kappy might have a better answer.
I didn't have a pure java app to try, but I did notice that if it is a .jar, you can set the default Open With to the 32-bit Jar Launcher. The other thing I was thinking was similar to Kappy's suggestion with Safari; just see if you can set it to open in 32-bit from the Get Info window.
Here is some info about java and Firefox:
The Java firefox uses is determined by the bottom window selection in Java Preferences.
The Java Safari uses is determined by the top window selection in Java Preferences.
Test your java here:
Try This, I think it might fix your problem:
Go - > Utilities -> java preferences
Click and Drag the
and Bring it over the 64 Bit Java
That should make it look like:
which should launch 32bit Java first.. then 64bit.
You can also choose to disable 64bit altogether though I find it better to just disable it.
You may have to restart your computer.
Let me know if this helps!