AppleScript: Display alert or restart when application terminates/crashes?
Quite easily with an Applescript:

 repeat
    tell application "System Events"
        if name of every process does not contain "Dropbox" then display alert "Dropbox has crashed!"
    end tell
    delay 5
end repeat


You could replace "display alert" with tell application 'Dropbox' to launch ..to have it automatically restart if you desire.

Paste this into Script Editor. save it, and then you could add the script to your user's startup items for future use. The timing there may be a bit tricky though if the script starts before Dropbox - adding a delay 10 or something before the first repeat would solve this.

For future use, anything involving System Events is a great place to look for solutions to these kinds of meta-system-wide tasks.