Generally airbag has no connection with the seat belt. Though airbags can hurt if you are not wearing seat belt, very often it is less than the damage caused by the impact of the accident. So that kind of a mechanism doesn't make much sense.
But, in some cars the passenger seat has sensors in it which measures the weight of the passenger. If the weight is below a certain point ie.if the occupant is not an adult, the airbag will not deploy because it can result in serious injuries to the young occupants. ABS
and seat belts are not interrelated.