Dobló Wine Bar
Dobló, which opened in 2010, was one of the first wine bars in Budapest. Being in the center of the Jewish Quarter (aka party district) means that more than half of the patrons are usually tourists, but here you don't need to worry about a rowdy stag party spoiling your fun. In fact, with a dimly lit, cozy interior and live music most evenings, Dobló is one of the more atmospheric wine bars in Budapest that also works well for a date night.
The wine menu features Hungarian-only vinos, of which 30 types are available by the glass, more by the bottle. When in doubt, ask the knowledgeable and friendly staff for a recommendation (they'll generally pour tastes too). Prices run €5-6 per glass. For an intimate experience, sit by the nooks upstairs.