If the mother alone is on the birth certificate, I believe she will be considered the sole decision maker for the child.
If the father is listed, it would be wise for the mother to get legal documentation that indicates that she has sole custody. She whould talk to a lawyer. This isn't just for the cruise, but it will help her in any number of things the come up in the future. The laws presume two involved parents so there may be other cases where the father has a say.
If he is not involved and not paying child support, it shouldn't be too hard. One approach is if he is not paying child support, to have the lawyer indicate that the mother will go after child support unless he signs an agreement giving your sole custody.
If he is paying support, he is probably very resentful as some men are and figures that not signing the letter is his way of getting 'revenge'. This one will be trickier, but if it can be demonstrated that he has made no effort to involve himself with the child, the court will usually look out for the child and grant all legal rights to the mother.
Issues can come up in the future regardin education, medical issues, etc. If the mother wants to move out of state at some point, she may have issues, unless she has clear legal custody. It would be a good idea to get the matter settled once and for all.