Some cruise lines are known for having more family-friendly rooms than others. Almost all cruise ships have adjoining room options so everyone can have a bit more space if your budget can handle it.
On Disney Cruise Line (pictured here), the family staterooms have a curtain that separates the kids from the adults and beds that fold out from the ceiling. They also offer Diaper Genies for free.
Both Holland America and Disney have bathtubs in every stateroom (others usually only have shower stalls unless you book a suite), which is helpful with babies.