Is this fluorescent light fixture a major electrical problem / fire hazard?
You didn't show the ballast, (the black boxy thing with all the wires coming out of it in the fixture) so my guess is that it went bad and some of the potting insulation around the coils of the ballast over heated and melted.  (Since I cannot see the whole fixture I'm just guessing about that.)  It looks like the black on the cover is a tar type residue but I really would not touch it.  It is sticky and a mess to get off.
I doubt anything was catching fire, but when it happened there would have be smoke and a really stinky smell.  If you smell the ballast you will see what I mean.  But back to the fire, I'm sure this has caused fires in a very few cases.
I would disconnect the power or turn the switch off.  The ballast has a thermal cut-off that would probably de-energize the ballast, but it still has power.
It can be fixed, but for a DIY'er it is easier to change the whole fixture.  To fix it you would have to wire in a new ballast and that would be 8 wires, where a new fixture would be only 3, a hot, neutral and ground.


If you do it yourself, make sure you turn the breaker off before taking the old fixture out.  Out of the ceiling will be either a black or red, a white and a green ground.  A new fixture would show you exactly where to put the wires.  If you have a friend that knows how to do this then get your friend to show you how so you can see what it takes.