And if you had a red shirt,
Then you'd be staying down.

Anyway, while Kirk was captain (as opposed to admiral) he only had, at peak, a crew of approximately 430 men and women. In Star Trek VII: Generations he commanded the NCC-1701-B for a bit which was designed for a crew of 770 but at that time it was not fully manned, much to his distress.

During his time as admiral he presumably had many more men but the song isn't titled The Grand Old Admiral Kirk.