If your children have valid passports, I would recommend using them instead of the birth certificates. They do have to have photo IDs along with birth certificates, and as the passport covers for both forms of ID, it's much easier to just use those. Same for adults, by the way. Definitely bring the birth certificate (not a hospital one, but the official raised-seal one from the county or state) for the baby. Photo ID is not necessary for an infant.
Are both of the legal parents traveling with the children? If any of them have a parent that is not coming with you, you must have a notarized statement from that parent giving permission to take the child out of the country. The specific cruise and destination should preferably be mentioned. Some countries, Mexico included, are stringent about that rule.