As with any item of value (I realize that a passport does not have a direct value), I would immediately call law enforcement.
At LGA, law enforcement left cell phones and other items around so see what people do with them and to catch thieves.
I would prefer not to be questioned by a potential thief in a "honey pot" sting.
Having said that, I would never return the passport to the individual. That is the State Department's job. Yes, it may cause some paperwork and the issuance of a new document with an endorsement that it replaces one that was lost & recovered, but it is also a protection for the passport holder if the passport was somehow improperly used.