The Upholstery Store is my go-to spot for quiet, serious conversations outside of the house. They only have wine, though, no cocktails or even beer, I don't think.
North Square's a good recommendation.
And another one on Washington Square: La Lanterna.
129 MacDougal St, New York, NY 10012
103 Waverly Place, New York, NY 10011
The Upholstery Store
713 Washington St, New York, NY 10014