Hi - we just travelled with an infant and young child on a Celebrity ship, and it was a disaster. We had such high hopes, but it was just really tough with the regimented eating schedules, etc.
We used private babysitting for 2-3 hours per night, and it saved us. There were no cribs in the group playroom, so I'm not sure where your little one would sack out.
If you're nervous about how your little one will cope, think hard about a stationary vacation as opposed to a cruise.