G. B. Shaw's 
90 Fulton St (Gold St / Williams St) 406-1380
Darts:
One dart board in the back of the place. Unfortunately it hangs above the table, so the darts are available only on the Monday league night. All other times it is strictly for decoration.
Overall impression:
Small place, mostly after work crowd.

John Street Bar & Grill  
17 John Street (Broadway & Nassau St) 349-3278
Darts:
This is a place to go to play darts. Almost every night there people playing.
Overall impression:
This pub is huge and offers 18 taps, very spacious, plush, walnut-stained, pine-and-brass-laden "old-style" beer room.
HAPPY HOUR:
Every Tuesday $2 beers and Thursday $10 all you can drink.

M. C. Exchange
28 Trinity Place (Rector St / Battery Tunnel) 509-0205

Metropolitan Improvement Co.
1 - 3 Madison St (Pearl St / St. James Place) 964-0422

O'Hara Restaurant & Pub
120 Cedar St (Church St / Greenwich St) 267-3032

Ryan's Sports Bar
46 Gold Street (Fulton Street / John Street) 385-6044
Darts:
The board is well hidden in the back of the bar. you will not see it unless you ask or see somebody shooting. 
Overall impression:
Excellent Irish bartenders that can pour a perfect pint of Murphy's stout. Regular bar food.
 HAPPY HOUR
Regular beer price is $3.

Square Rigger Pub
163 Front St (John St / So. St. Seaport) 422-0188
Darts: 
Two dart boards hanging on the wall. The management puts tables in front of them during lunch (11-4), so they are not usable during the day.
Overall impression:
Little too touristy. 
HAPPY HOUR
They always some king of food special on the white board.

The Beekman Pub
15 - 17 Beekman St (Williams St / Nassau St) 732-7333

White Horse Tavern
25 Bridge St (Whitehall St / Broadway) 952-9722

Dart Bars in NYC by NEIGHBORHOOD