I found this interesting. I emailed Stats, Inc. and they replied with this awesome information about what teams have drafted the most players that are currently on Major League Baseball rosters:

Of the 1,070 players who have played in the majors this season (or have been on the major league disabled list) here are the teams with the most and fewest drafted and SIGNED players.

Most Drafted and Signed Players in Majors:

Toronto Blue Jays 51

Atlanta Braves 51

Oakland Athletics 48

Arizona Diamondbacks 42 (includes John Patterson who they did not draft, but did sign after Montreal failed to offer contract in 1996)

Seattle Mariners 41

Fewest Drafted and Signed Players in Majors:

Milwaukee Brewers 15

Cincinnati Reds 22

San Diego Padres 22

St. Louis Cardinals 28

Tampa Bay Devil Rays 28

A player like J.D. Drew, who was originally drafted by Philadelphia but never signed, would be credited to the St. Louis Cardinals who drafted and signed him in the following season.