Starting in December 2015 the Java Web Start client and applet were abandoned. Over time Java's code signing requirements became increasingly difficult, and finally these were abandoned. The new clients include java with them, and have a client for Windows, one for Mac, and one raw .jar file for Linux systems. Instructions for each are below.
TBD
To enable KGS on a Mac, go to system preferences, open Security and Privacy, click on Allow apps downloaded from App Store and Identified developers. Then open the dmg file again. If it doesn't run, open this same screen and you will see a notification saying an app was blocked from running. Click allow and it will work. You only need to do this the first time you run the program.
For Linux you need to use the raw .jar file. Go to the Java download page, and download and install (or install the Java that came with your Linux distribution). Then download the .jar file, and run from the shell with:
$ java -jar cgoban.jar
The "old" CGoban download page.