This is New York City - everyone takes their kids out to eat! As long as you are sensitive about choosing restaurants, there is no reason you can't go out to eat with your baby. There are many restaurants that are obviously child friendly (look in the window - lots of kids!). But there are also restaurants that you might not normally think of as "childrens" eateries, that can be very child friendly at earlier dinner hours before their "older" crowd comes in and they are less crowded. Go out to eat with friends or family. If your child does become fussy, you can take turns eating and walking around (inside or outside) with the baby. This is a time honored tradition when eating with babies! You'll also get to know your baby. She may be less fussy in a quieter smaller restaurant where there is less stimulation. Or if that doesn't seem to make a difference, choose larger less intimate places where the baby's noise won't be disruptive. As long as you use some judgment you'll have a great time and won't be a problem for anyone.
Also remember, children who go to restaurants regularly with their parents, also have the opportunity to learn to enjoy something you enjoy, learn to eat different foods (even if only through your modeling in the beginning) - and importantly, learn how to eat and behave in a restaurant, since hopefully you will be teaching and expecting age appropriate restaurant manners all along the way.
So go and enjoy!