Hellobee Boards

Do you have a specific neighborhood you will be in or want to stay in? That might help narrow it down a bit...

Jacob's Pickle, Bubby's, and Pizza Sirenetta (fancy pizza, not like jumbo slice, haha) are the first couple that I can think of off that top of my head

I stay in Brooklyn mostly but I love Carmine's (Italian) huge servings (family style). My splurge place is Plataforma. Brazilian steakhouse. We usually go at an off time (in between lunch and dinner) so it's not crowded. They have the best kids goodie bags ( lots of random things, sunglasses, coloring book, plastic tops).

Sarabeth's is an NYC staple for kids.
The Smith (it's a little loud which makes conversation challenging, but it's good for masking the sound of kids crying )
I've heard great things about Hardings for kids but have never been.

I agree with @Mrs. Bee: that most places will probably be fine, especially if you're going to dinner on the earlier side (5/6pm). I usually just call and ask if they have highchairs and if they do, that's a sign to me that they are ok with kids.

If you like cool tapas type places, Stanton Social and Beauty & Essex are both good and have been around a long time. Unique stuff like tomato soup and grilled cheese "dumplings". I would just call and ask about highchairs / kids. I've only ever been there later at night.
Also, The Dutch is phenomenal and a hip spot, again, I'd call and ask about kids.

Oh, great area (I thought that's where you guys might be!). I live on the UWS, so it's just a quick straight shot south a few blocks. I can probably really only recommend places in this neighborhood since I rarely leave it, haha...All these are located between West 96th Street and West 59th Street.

Mermaid Inn (seafood, 'cool spot', excellent food, believe they have limited outdoor seating)

Nice Matin (French, outdoor seating)

Isabella's (American nouveau, classic UWS, outdoor seating)

The Smith at Lincoln Center (American classics, fun atmosphere, outdoor seating)

Landmarc in the Time Warner Center (overlooks Central Park, American classics)

Red Farm (updated Chinese, hard to get into)

Seraphina's (Classic NYC, Italian, has limited outdoor seating)

Playa Betty's (California cuisine, very family-friendly)

@caterw: OH if you're by Columbia there are lots of places nearby that are great with kids! Friedman's, Flat Top, Pisticci, and Community are all delicious and in Morningside Heights and very, very friendly with kiddos.