I would do a car just because of the baby stuff you will need and the flexibility on time.
I would also find a hotel with parking and not move my car again until I left.
I am super anal retentive about car seats but have to say I would use cabs and just hold my baby if staying in the city itself. I think the average speed in NY city is 5mph
I have done the train without kids and flown without kids and I prefer the train. Its not quicker to fly when you factor in parking at airport, security, flight itself, waiting for bags at other end. Well, maybe its 30 mins quicker but still prefer train.