Yes sure, I can connect to any database that you have a JDBC driver for. I have much experience doing this. Its easy to do, MySQL comes with a JDBC driver.
*edit*
Though as for Java applets, let me know what you have in mind. I have some Swing (GUI) experience which is okay for running desktop tool stuff (its stable). I have a lot of experience running up browser interfaces using DHTML run up by Servlets. Its easy to install on your server as well. But if you want to run a Swing GUI inside a browser, this used to be a bit buggy about 5 years ago last time I tried anything complicated with it. About the time of Java 4.0 I think it was. I don't have any experience of the new native Widget GUI stuff, but I'm not sure if thats properly portable yet.
Best regards Paul