Unless you have a paid support contract, you can only update Java 6 to update version 45, and they reference 81.
If you need to be able to use Banner and Brio on campus, you are stuck at Java 6. Please use NoScript or a plugin along those lines that disables Java except on sites that you tell it you trust.
Java 7 update 80 is still available if you need to stick with Java 7
Java 8 update 91 is available for download from their site if you have no software restricting the version you use.