I would probably go with both: hardware cloth (predator proofing) plus tarps (wind/rain/snow). The tarp I would use over the top and some of the sides, but I would also leave it off some sides for ventilation. How many open sides? Depends on the climate. In Florida, I might cover just the roof. In Michigan I might cover the roof plus three and a half of the four sides (maybe just leave the door un-tarped.) Other places, something in between.