We went on a family cruise in April, which included our then 15 month old son. We brought our City Mini GT.
We used it around the ship, though it took some finagling to get it in and out of our stateroom door.
As far as restaurants, we found that they had "parking areas" outside of the restaurant, which leads me to believe the strollers were not welcome inside. We did not try to bring a stroller inside, and I can only imagine that it would have been in the way. It's pretty tight.
Sometimes, we did "park" the stroller out of the way on the main deck so that we could walk around and no one ever touched it or anything.
You would need to use the elevators, but honestly if the line was too long, DH and I just carried the stroller up or down the stairs.
We definitely needed it for our port excursion.
Overall, given the choice, I would have totally done it the exact same way. Bringing our stroller was very useful to us.