Closed-loop cruises require a birth certificate for travel but do not require a passport. These cruises start and end at the same port within United States territory.
Open-loop cruises require a passport for travel. These cruises have different starting and ending locations, which might be in different countries.
Any travel internationally requires a passport for travel to and from the United States. This requirement also applies to children.
To enter some foreign countries such as Mexico, a birth certificate might be sufficient to enter the country, but you will need a passport or other approved document such as a trusted traveler card to re-enter the U.S.
Children younger than 16 years old traveling by land or by sea can travel to Western Hemisphere countries such as Mexico and Canada with a U.S. birth certificate.