Start on Stone Street, then widen out
If you only have one evening and you want a drink downtown, walk to Stone Street. It's a narrow cobblestone block between Broad and Hanover, closed to cars, with bars and restaurants that all put tables out along the middle of it. From our door on John Street it's about eight minutes on foot, straight south. Nowhere else in the Financial District gives you that many options within thirty steps of each other.
The rhythm down here is worth understanding before you go. This is an office neighborhood, so happy hour is a real event Monday through Thursday: the block goes from empty to full between about 4:45 and 5:30, stays loud until eight, and then thins out fast. By ten on a weeknight you can have a bar to yourself. Weekends flip it, because the office crowd is gone and the people out are mostly visitors, which means shorter waits but fewer deals running.
One caveat before the list. Happy hour windows and prices in New York change constantly, so treat the specifics below as a starting point and check the bar's own page or ask when you sit down. Everything here is a short walk from 17 John Street, and none of it needs a reservation on a weeknight if you're there before six.