If you go to the NYC subforum you will see a top questions section which includes information about how to get from the airports. The fare is already set.
I am not sure what safety issues you are concerned about. Of course you could end up dead in a car accident. If you are concerned about being murdered or robbed at gunpoint by the taxi driver, that is just silly.
NYC taxi drivers are not allowed to negotiate fares except outside NYC - where most do not want to go. Within the city they -and you - are required to use the fare on the meter - plus any tolls and the usual 20% tip. The fare from JFK to Manhattan is fixed at $53 (plus $4.50 if during rush hours: 4-8 pm) plus the $6 toll if bridge or tunnel to Manhattan and again the usual 20% tip. So a total of either $70 or $75.
You can pay with cash or credit card and all the details are on an interactive screen in the back of the cab.Edited: 16 May 2018, 04:15
Yes, from JFK airport a Yellow cab from the dispatcher line is safe, easy and a fixed rate to anywhere in Manhattan. Use only the dispatcher line. Ignore any touts offering (illegal) rides. nyc.gov/html/…AirportTrips
Possible exceptions would be 1) groups of more than 4/5 adults or those with excessive luggage (although splitting up into two cabs may be the easiest and reasonably priced option) and 2) anyone needing a child safety seat - you'd want to arrange a car service that can supply one. (Both cabs and car services are exempt from the requirement. so it has to be requested from a car service.)