P&O have free places for children generally outside of school holidays. However the kids club does not start until age 2 so you should wait 6 months, although there is a supervised night nursery in the evening until 2am for all, See this link
I think Costa and MSC are offering free child places. Of course children in nappies are not allowed in the pools on the ships - how unhygenic.
P&O's new ship Ventura has a special under 2's parent and child play area and they will also have noddy onboar.
Ocean Village also has good child prices and good facilities for children and offer babysitting services, it can also depend on what type of cruise you want - informal or formal.