From the minor league baseball website:
AAA - 46 teams
AA - 30 teams
A (high) - 30 teams
A (short season) - 22
Rookie - 82 teams
210 teams x 25 players = 5,250
There will be some overlap between short season and rookie players. Assume that to be 500 players? Making the total minor league players somewhere around 4,700 each year.