First. let me say that, IMHO, anyone who is using Java in a browser to play the chess computer is taking foolish security risks. There are all kinds of nasty, nasty zero-day exploits that can hit you via Java-In-The-Browser. I'd recommend that you simply wait until V3 is released (whenever that might be). The V3 computer won't require Java at all.
However, if you want to throw caution to the wind (I DO NOT recommend it), Post #20 in this thread might give you some information:
http://www.chess.com/forum/view/help-support/java-development-toolkit
Just recently I lost the ability to be able to finish the game against the chess.com computer. First google chrome decided I needed Java. It's there. Then I tried Internet explorer. It worked until it decided there was a security issue. Anyone else have this problem? What is the solution?