I have a project I maintain in Java that runs primarily as an applet. I am finding that Applets are simply unpractical and dangerous for browser games as most end-users ignorant
Hmm.. GWT compiles java to JavaScript you can use it.
Another possibility would be using Rhino
Or just expose REST services and create an HTML5 web application using it.
Edit: BTW search in the forum. I found some questions like this one.
I found this tool called PlayN which may be what you're looking for. I've never used it before, but it looks promising. Of course, it may require rewriting your Java source to use their API, which means, as others have said, you're probably best off just manually porting to JavaScript.