The exposed fiberglass insulation is fire code in most states. There can be no paper on any insulation that is not covered with wall board, in case of a fire.
Fiberglass insulation will not degrade unless exposed to water. The fiber could become airborne if a vent is blowing on it or it is disturbed in some way. It can be covered with a plastic vapor barrier to protect your home air quality.
It is important to note that if your house has a vapor barrier on the outside of the house and it is not semi-permeable like a Tyvek or Typar housewrap, you could be trapping vapor between your walls. An example of a vapor barrier that is not semi-permeable is black felt paper, also referred to as tar paper. It would be smart to investigate what type of vapor barrier exists on the outside of your house (if any) before putting up the plastic.
The plastic can be bought at any big box store. It should be clear and at least 4 millimeters thick. If the plastic vapor barrier is sealed properly, this will also help with air infiltration into the house, lowering energy costs.