Has a baby ever been born on a submarine in active service?

Not likely. Women on submarines is a very rare and recent thing.
http://thehill.com/policy/defense/219685-enlisted-women-to-begin-serving-on-submarines
http://www.bbc.com/news/uk-27278588

Believe it or not, the answer is yes! During WWII, the USS Crevalle (SS-291) was evacuating refugees from the Philippines, and one of her passengers gave birth on the trip to Darwin, Australia.

"Crevalle (SS-291) During a battle in the Pacific, a baby is born aboard the submarine USS Crevalle. He is the son of a Filipino woman being evacuated from the besieged islands. Somewhere in the Philippines today, a young man answers to the name of Elma Walker Crevalle Talcaban. How he received his name provides an absorbing story of wartime heroism aboard a U.S. submarine."

The story is told in an episode of the TV series "The Silent Service."

https://en.wikipedia.org/wiki/USS_Crevalle_(SS-291)