Try opening the Java settings menu from the control panel and deleting the certificates you have there for RuneScape/Jagex. It should redownload them. Control Panel->Java->Security->Certificates
If you have no certificates there, then another thing you can try is deleting the RuneScape cache. It stores it at C:\.jagex_cache_32 You can just delete the directory and it will redownload everything.
If neither of those work, there is always the
Official RuneScape client. I prefer it to playing in a browser. It comes with it's own bundled version of Java so it should always work.