I have visited Napoli many times over the last 20 years or so and I think just this year that an effort is being made to clean up a lot of the problems. Taxis are always a breed unto themselves. There is a bus that will take you from the airport to the train station. I'm not sure of the times, but it's a very short ride.

Slightly off topic, but maybe useful for tourists, is one of the other long-standing scam in Napoli which is at the train station. State trains arrive at the main station, but the popular (but filthy and run-down) trains to Pompei and Sorrento are a private railway and operate from a different area of the station. Of course the signs are not easy for a tourist to understand. In other parts of Italy the signs would read "Trains to Sorrento" but here the signs read "Circumvesuviana railway". How does a tourist know this? If you look lost a friendly man in an official blue shirt will help you find your way, even helping with the luggage then demand an extortionate tip. The whole problem has been going on for as long as I've been visiting Napoli, is still going on, and could be solved by a few simple signs.