Try Forgione on Reade Street in Tribeca. The food is fantastic and can be paired with an extensive wine list. Reasonably priced for the quality of food and service. Exposed brick and low lighting make this place an excellent choice for a date. Their is also outdoor seating on Reade Street if that suits your fancy.
Information on how much you want to spend would be helpful. Two places that come to mind . . .
Terroir Tribeca - New, much bigger than the original, friendly service, only had a few bar snacks so can't really speak to the food.
Locanda Verde - Still very popular (and difficult to get a reservation at a prime time), solid food though not the most adventurous
413 E 12th St, New York, NY 10009
377 Greenwich St, New York, NY 10013