Some of the migrants swam from the boat to shore, while others jumped out onto the beach after the boat landed.
"The boat was unseaworthy and grossly overloaded," Coast Guard Petty Officer Jennifer Johnson said.
It was not immediately clear exactly how many people were aboard or if any were children, Coast Guard spokesman Dana Warr said. He said one body washed up on the beach, and the Coast Guard and local authorities were searching for other possible migrants.
The group had been on the boat for several days, said Border Patrol spokesman Victor Colon. Immigration officials planned to interview the migrants Wednesday.
Haitians who illegally make it into the United States are generally sent back. Most Cubans who reach American soil are allowed to stay under U.S. policy. Last year, Coast Guard agents patrolling the waters of South Carolina, Florida and the Caribbean stopped 6,061 migrants, 769 of them from Haiti.