Greenwich village is a fun and great neighborhood for a date. Up & Up, which is ironically subterranean, offers a choice venue for a drinks date. The pseudo speakeasy takes reservations and the space is large enough that you are likely to be able to get one without having to book weeks out.
The atmosphere is demure and discreet, the drinks are good and small bites can suffice if you need something to nosh on between sips. The staff is pleasant and the prices are reasonable. This is a place to bookmark for your social calendar.