When Firefox drops Java applet support, how will I run old Java applets?

The Java Developers Kit (JDK) has a utility that is designed for debugging applets. The tool is called appletviewer and is run like this:

<PATH_TO_JDK>\bin\appletviewer.exe <url_to_page_with_applet>

I'm not sure how viable of an option this is, but it should allow you to see many applets without using a browser.

According this post some browsers will extend support for NPAPI for a while:

What are the timelines for browsers' plugin support?

Individual browser vendors have been updating their plans regularly. Here's a snapshot of what some browser vendors have stated as of today:

Microsoft Internet Explorer (IE)

Microsoft has indicated that they intend to continue to offer plug-in support in IE.

Microsoft Edge

Microsoft Edge was released in Windows 10 without Browser Helper Object (BHO, aka. plugin) support. Microsoft has no plans to add plugin support to Edge.

Mozilla Firefox Extended Support Release (ESR)

Mozilla indicated in early 2016 that Firefox ESR 52 will be the last version to offer NPAPI (and JRE) support. Firefox ESR 52 is targeted for release in April 2017 and will be supported until May 2018.

Mozilla Firefox Rapid Release

Mozilla intends to remove NPAPI support from the Firefox 53 Rapid Release version targeted for release in April 2017.

Apple Safari for Mac OS

Safari offers Internet plug-in support for Mac OS users. Apple has not made any statements about deprecating plugin support for Mac OS users.

Alphabet (Google) Chrome for Windows

Chrome offered support for plugins until version 45, released in September 2015. They removed NPAPI support in later Chrome releases.