For anyone reading this in 2017 or later when no modern browser supports java applets, here's what I did to get this to work on Ubuntu 16.
Install icedtea to get the java applet plugin:
sudo apt-get install icedtea-8-plugin
Download firefox 51 (last version before java was removed)
Extract to a specific location ($HOME/java/firefox, for example)
Disconnect from the internet temporarily (otherwise firefox will update itself to a post java version immediately)
run firefox --no-remote
Go to about:preferences#advanced and select 'Never check for updates'. You can now re-enable the internet.
After launching this browser, do not use firefox for any normal local web browsing as you will be using an old version which is unsafe. This is because when you launch firefox it will try to attach to an existing process, and so will pick up the old one. If you normally use firefox (or any of its derivatives such as iceweasel), I would suggest running the old one with $HOME set to something other than the usual, or maybe even as a different user to avoid a conflict in settings.