Just because Java applets aren't popular doesn't mean they aren't powerful or worthy of gaming. In fact I am focused on Java game development at the moment; I like the high-level nature of the language, its cross-platform-ness (as compared to C#), the JOGL library which provides a very C-like OpenGL experience for the Java platform (or its alternative, LWJGL, or an engine like jMonkeyEngine), and its speed (obviously slower than C++, but considerably faster than scripting languages like Python).
With Java you can deploy your application as an Applet to run in the browser and even provide a JNLP alternative for those who prefer installing something.
You can take advantage of all the tools that have been created for Java; the wonderful Eclipse IDE (or any of the other IDEs out there), testing tools, continuous integration, the huge community of Java programmers, etc.
The statistic for what percentage of computers Java is installed in varies (and the version varies even more) but I've seen pretty high numbers for Java market penetration. And the installation process remains pretty simple, so that if any users don't yet have Java installed, they can be up and ready to run your game in very little time.
I personally feel less hesitant to install the Java plugin than the Unity one. In fact, I am yet to install the Unity game plugin on my computer; I just don't quite trust it. Java is proven; it had a rocky beginning but now, many years later, it is stable and fast and, well, it continues to have some security issues (but so does Flash) but for the most part it's trustworthy. I will continue to avoid Unity web games until the Unity web plugin is a little more widespread and proven.
So I absolutely recommend Java, whether through an applet or through Java Web Start or an executable JAR.
P.S. The world's most popular free online MMO, as recognized by the Guinness World Records, is RuneScape, which runs as a Java applet and recently has a desktop version. It supports both DirectX and OpenGL as well as a software 3D renderer, and has approximately 10 million active accounts.