<p>I have an third-party applet that requires JRE v1.5_12 to work correctly.  THe user is installing JRE v1.6.07 or better.  It used to be with 1.5 and below, that I could have multiple JRE's on the machine and specify which one to use - but with 1.6 that apepars to be broken.  How do I tell the browser I want to use v1.5_12 instead of the latest one installed?</p>

For security reasons, you can no longer force it to use older JRE's. Say release 12 has a huge security hole, and everyone installs release 13 to patch it. Evil java applets could just say "run with release 12 please" and then carry out their exploits, rendering the patches useless.
Most likely you have some code with security holes that the newer JRE is blocking, because it would cause a security risk. Fix your code, should be pretty minor changes, then you wont have to worry.
See
this page
for more info on the change.