UPDATE:
I haven't used Babylon since it became non-Firefox-friendly but today I tried something which worked. Here is what I did:
I have installed the latest version (Babylon 10) on my PC while Firefox window was open. As soon as the installation was completed, a page was opened in Firefox which asked me if I wanted to install "Babylon Translation Activation Tool". I clicked "Yes" and it installed that extension on Firefox. After I restarted Firefox, I tested Babylon on Firefox and it worked!
Here is a screeny of that Firefox extension. It is weird that when I search for that extension in Firefox extensions database, it cannot be found. Hope that helps.

I found the following post belonging to user "wimyogya" while searching the web for a clue:
Babylon Support earlier informed me
that at this moment they do not
support IE9, Chrome 10beta and Firefox
4. They recommend temporary browser downgrade.
Later I got the answers to my
additional questions:
- Our developing department is already working to make Babylon 9
compatible with IE9, Chrome 10 and
Firefox 4, and it should take about
3-4 weeks.
- Babylon Company is making the adaptations.
- Only users of Babylon 9 and users who purchased Babylon 8 in grace
period, from the 1st of December of
2010, will be eligible to get the
update.
Seems my Babylon 7 becomes pretty
useless :(
Too bad!