There is no doubt that pregnant women got on ships that were headed to the United States.
There is no doubt that some of them had their babies on the ship.
I tend to doubt the story unless I can find some relatively contemporary evidence of it. The infant listed on the manifest as a newborn is a prime example. Stories of the baby being born on the ship make for nice dramatic stories and, when it actually happened, there is no doubt that it would have been a dramatic birth and the baby was fortunate to have survived.
But try and find something that back up that story.
Just like you would anything else.