Google Chrome Java Workaround
The best option I'm aware of is Java Web Start (tutorial), which doesn't run in browser but does allow users to easily run a Java application you serve from a website. Oracle provides some documentation on how to migrate to JWS, but the gist is to simply replace the <applet> tags with a link to a .jnlp file with the same information the <applet> tag had. You may have to tweak Chrome's application defaults as well.
There is also the Google Web Toolkit, which isn't for applets, but does let you write Java that compiles into JavaScript.
If your applets are more integrated into the browser, unfortunately you may not be able to keep using Java. Chromium does offer some alternatives on their NPAPI deprecation page, but they're not pretty. Options include HTML5/WebGL, WebRTC, NaCL, or Chrome Extensions. The Chrome Experiments site is worth browsing if migrating off of Java becomes your best option.
Oracle's official recommendation, however, is to avoid Chrome. That's pretty heavy-handed, and won't work for most website owners, but it's worth mentioning.