Pegu Club would be my pick. Classy, but still relaxed and fun. It can get pretty loud and crowded by the end of the night, but before dinner, it isn't too bad.
For somewhere more speakeasy-ish, try Little Branch or Employees Only. LB opens at 7pm. EO at 6pm.
EO is more racuous and crowded, and the drinks aren't quite up to the level of LB or Pegu, but the decor is very sharp looking (art deco flourishes everywhere).
22 Seventh Ave South, New York, NY 10014
510 Hudson St, New York, NY 10014
77 W Houston St, New York, NY 10012