Applets aren't obsolete, but the current version of the JDK that ships with IE *is*.
Microsoft has no interest in seeing
Java thrive, and this is one of the reasons why they don't ship the "latest" version of the JDK with their browser. But, this in itself isn't that much of a problem - think of all the web-based plug-ins that require the user to d/l the plug-in first before they can be used.
If you limit yourself to the APIs that were supported in the early version of the JDK (1.1.4), then your applet will run fine in "stock" versions of IE or netscape. But you shouldn't limit yourself to this API, because they are old and frankly, not very robust. Download the latest JDK 1.4.1 plug-in from Sun, and use the modern Swing APIs for your GUI work.
Here's Sun's plug-in page:
http://java.sun.com/products/plugin/index.html