Stamina is not like game shape. The only way to increase stamina is by training it. The difference with this training compared to training something like Jump Shot, is that minutes do not come in to play. If you train any of the Team trainings (eg. Game Shape, Stamina etc) then it will train EVERYBODY on your team regardless of how many minutes they play that week.
However, stamina does drop slowly for ALL players. This is not affected by how many minutes you play, it is just random and out of your control. So expect some drops in some of your players for stamina.
On another note, training stamina is good during the All Star Break and Off Season when there are only 2 games a week. Stamina also trains relatively fast.