Yes, a letter is required for travel to Mexico, and strongly suggested all other times.
From the Carnival Web site (Carnival Minor Policies
"When traveling with a minor and both parents/legal guardians are not cruising, we strongly recommend bringing an original signed letter from the absent parent/legal guardian authorizing the minor to travel with you. This will expedite processing by the Department of Homeland Security. Please note that a notarized letter to this effect is required if debarking with children in Mexico."