New York’s main airports are JFK, La Guardia and Newark. These are well connected by public transport to the city centre. Buses and Trains are a good option, but after a long journey with a toddler a taxi might be more appealing as you will need to change trains if you opt for the airport trains. Shuttle buses from the airport to your hotel are also available but the problem with these is there are no car safety seat. For information on getting from the three main airports to Manhattan see: http://nymag.com/guides/cheap/airporttravel/.
Taxis in New York are exempt from laws on car safety seats and the option is yours to carry your little one on your lap however taxi rides in New York can be bumpy as the drivers tend to be a little over zealous. Using a car safety seat is always safer and there are options. You can either bring your car seat with you from home or you could pre book a taxi with a car safety seat from many companies including KidCar New York (https://www.kidcarny.com/) (provides cars and mini vans). The company arranges airport transfers and can also take you on city tours.