Unable to run Java-based games outside of IE

The problem occurs during the usage of either the official Java applet
provided by RuneScape's game company, or by any alternate fan-based client
for the game of RuneScape, known as SwiftSwitch.

Both of these programs run perfectly, until the loading page is reached, for
the Java applet to begin instalation.

Yet, this same game, reached via the website using Internet Explorer 6,
works perfectly fine with Sun Java installed.

How can I allow Java to run, outside of IE6, on their respective
programs/clients?