I am looking for a way to save a couple of files created with JavaScript on the clients computer. I want him to be able to choose a folder (similar to open/save folder dialog),
It sure is possible.
One way is to use a signed applet, but if your users are willing to upgrade to Java 1.6.0_10+, a signed applet is not needed. Since 1.6.0_10 (the Next Generation Java Plug-In), JNLP API services are available to embedded applets. Here is a demo. of the JNLP file services.
I see the other reply has already covered the applet/JS interaction.