What to do if the garage door at my apartment was damaged by an unknown person?
No, they are not negligent for not having security cameras for this purpose. However, they can only hold you liable for  this damage if they can show or prove that you, or a guest of yours, did it; that  is, you would not be liable for damage done by some third party not under your control. If you do not pay, they will either have to sue you for the money (and prove in court by a preponderance of the evidence, or "more likely than not," that you did the damage) or else take it from your security deposit; if they do the latter, you could then sue them for its return, having to prove in court that it was not you who did this. You could use witness testimony (yours or other persons') about your whereabouts at the time or that you didn't do this, as well as other evidence, such as photographic evidence that your car has no matching damage.