NYC is costly, but not as costly as people think. The lifestyle is also different.
1. Most of my life in NY (pre children), we had no car, no car insurance, no parking, and rare taxis (and before our current mayor, subways worked very, very well, I rarely considered taxis.)
2. Property taxes are comparatively low to other states
3. Access to culture is widespread and cheaper than people think -- there are subsidized tickets for students, etc. Museums are plentiful and often free. Libraries are literally everywhere.
4. Access to jobs is great and you dont crush your soul with hour-long drives. And when you lose/change your job, your next job is also often in the city center, so you dont have to move homes.
I live in DC suburbs now and we have one car. It is hard getting places w/o a car and the train system is good but does not cover as much of the metro as one wishes, and there are last mile issues. Property taxes are way, way higher both both homes and other property (e.g., $800/yr on my Kia minivan), but there isnt a city income tax.