It’s the passenger in the aisle seat who is most often injured by baggage falling from an overhead bin. The injuries can be serious and can include mild traumatic brain injury.
If the baggage falls and injures a passenger who is travelling internationally, then the Montreal Convention or Warsaw Conventions apply. The conventions are international treaties that make the airlines automatically liable for any injury to the passenger that resulted from an "accident." An "accident" is defined as an unusual or unexpected event that is external to the passenger. Under certain circumstances, being injured by falling baggage may well qualify.
The conventions apply even if the flight was entirely domestic, as long as the passenger had an international destination somewhere on his itinerary.
What if the flight on which the injury occurred was domestic and there was no international travel involved? Then it’s trickier. The passenger must prove that the airline was negligent before the airline can be held liable. For example, the passenger must prove that a flight attendant was careless in opening a baggage compartment and allowing the object to fall out. Or, the passenger must prove that the bag fell out when a fellow passenger opened the compartment because a flight attendant stowed the bag improperly.