The solution is increased enforcement of the laws that make these illegal.
The incentives aren't difficult: Create an anonymous tip line to report illegal AirBnB operations that violate zoning regulations. Have submitters provide the street address and URL of the AirBnB to make it easy. The person operating the AirBnB (or owner of the house) gets fined. The person submitting the tip about an illegal AirBnB receives 25% of the fine.
It wouldn't take long for illegal AirBnBs to close up shop when they realize there are negative repercussions for what they're doing. Keep it anonymous to avoid retribution.
I could even see bargain hunters looking for illegal AirBnBs to stay in, then reporting their hosts to collect their share of the resulting fine to recover some of the cost. The incentives are deeply stacked against the illegal AirBnB operators in this scenario.
Of course, this only works in areas where it's illegal to run temporary rentals. I expect we'll see more of those regulations as the problems with short-term AirBnBs in residential areas become more apparent.