Java applets in a browser on Mavericks?
I´m using OSX 10.9.2 and Java (version 7 update 55) with Firefox (Nightly) and latest Safari and it works. It is right, that Java will not be shown in Safari extensions but I could find it under Firefox plugins.

I suggest that you uninstall Java, check that everything will be wiped off and reinstall Java. Normally Java will be enabled in system settings by default.

Update: If these doesn´t help please create another user and logon into a fresh user profile. Do you´ve got the same bug in Java system preferences?