I first tried to use the OpenJDK project but could not get it to work.
I ended up having to download an RPM of the official JRE from sun.
I then installed it (replace the file name below with the name of the file you downloaded).
yum install ./jre-6u30-linux-i586.rpm
Then created the following directory if it does not already exist.
mkdir -p ~/.mozilla/plugins/
Next setup a symlink. The following will work for the 32-bit version.
cd ~/.mozilla/plugins/ ln -s /usr/java/jre1.6.0_30/lib/i386/libnpjp2.so libnpjp2.so
If you have the 64-bit version do the following.
cd ~/.mozilla/plugins/ ln -s /usr/java/jre1.6.0_30/lib/amd64/libnpjp2.so libnpjp2.so
Now restart your browser.
This works for both FireFox and Chrome.