When they say crashed browser are we talking about an actual bug in firefox or a plugin that results in a crash?
Debugging is possible, but it assumes a developer experience.
ulimit -c to make sure your system will actually create a core dump with the application crashes. Then use GDB (*nix) to debug. You will probably want to download the source code for the browser. You may want to recompile your browser with the debug symbols.
Unless the question is for a developer position though I doubt you would be expected to be able to do this.
Are you sure they are asking how to debug a crashed browser, or are they talking about finding a network issue which should be very easy to diagnose from a command line. In that case you would be using something like
traceroute, and so on.