I think serious applets have always been rather restricted to intranet and extranet applications rather than websites. As Flash creators became more common a few years ago those led to a decline in trivial/gimmick applets as site authors flocked to the new tool to create them which didn't mean they had to learn a programming language.
Applets of course are the one area where Java didn't bring the expected results. This has always been the case and I don't see that changing. If anything the prevalence of other technologies (like JavaWebstart) will make them even less visible. Personally I don't think that's any great loss
I still fully explain applets in my Java books. (I also cover servlets!) Even though applets are not as important or as widely used as was expected in the early days of Java, they are still an important part of Java. Applets are part of the reason for the early success of Java.
In my view no one can honestly call themselves a Java programmer unless he/she can write an applet. Also, I like applets from a conceptual point of view. They are great for doing simple things.
As to their future? First, they are here to stay. However, unless something changes, their use will continue to be limited. That said, things change quickly in Java World. Nothing would surprise me.
Also, I do not cover Web Start in my book.
For my latest books on Java, including my Java Programming Cookbook, see HerbSchildt.com