We lived in New York for a few years, but came back 6 years ago, so I am horribly out of date (things change quickly there).
Expensive? Yes, for some things. e.g. rent, food shopping, bars. But other things are more affordable than here: electrical goods, public transport, eating out (at least cheap places are better quality than here. There are Very expensive places too!)
Families: Tons to do, and a lot of them are free, so you'd have a ball. Great zoos, botanical gardens, parks, playparks, bookshops, libraries, outdoor swimming pools, concerts in the park. It's quite an outside sort of lifestyle, because people don't tend to hang around in their pokey apartments, and hardly anyone in the city has a garden.
Where to live: a lot of families live in the city, but Brooklyn, Queens, Staten Island, rather than Manhattan. People also live in New Jersey or Connecticut, which have lots of nice towns, although this involves a bit of a commute. If you want a house and garden though, this is where to go.
We lived in Brooklyn, in a very family friendly part. Finding a good apartment is the hardest thing, and you have to accept that you will not have a lot of room, but then the good thing about living somewhere temporarily is you don't tend to accumulate things. In fact, a lot of people move in and out of New York, so there are always stoop sales (like car boots but on people's stoops/steps) with people selling on things like crockery, hoovers, bits of furniture etc. You can kit out your apartment pretty cheaply.
There's a big cafe culture - you can hang out in them with young kids quite easily. And there are lots of small cinemas and libraries in each neighbourhood. Most people don't have cars (parking is a nightmare and expensive in Manhattan) so there tends to be lots of local activities that you can access by bus or subway.
Supermarkets are small and pretty awful to use (cramped), and don't have anything like the things we have, but eating out or ordering in is so easy and varied that you stop minding so much. Take out is the New york equivalent of ready meals.
Don't know what your DH does, but the only thing I can think that might be a problem would be the visa situation. If it is big companies that he's applying to, they'd probably sort all that out, but it can be a bit complicated and drawn out. You wouldn't get a work visa, only a spouse's residency visa, but I'm guessing this isn't an issue with young children.
I really enjoyed our time in NYC. It was quite exhausting though - the pace of life is fast, and employees work very hard with very little time off. Being a housewife is tough too if you don't have spare cash e.g. dragging your laundry to the launderette (you can get service washes including delivery, but it costs), and shopping in the awful supermarkets with young kids in tow and having to slog home on foot or the bus (all prams have to be collapsed on the bus!).
But the fun stuff made it all worth while - great for arts, music, cocktails, vibrancy, full of characters, people from all over the world, and all the usual iconic New York stuff like the museums and tourist attractions. Feel free to ask me anything, but there are probably lots of people who live there now or are from there who can give you better info than me.
hth