Florida's Gov. Charlie Crist declared a state of emergency and said Fay threatened the state with a "major disaster." Forecasters said Fay could bring hurricane-force winds to the Florida Keys as soon as Monday.
The U.S. National Hurricane Center in Miami said that on Saturday night the storm was located about 60 miles southwest of Guantanamo, Cuba. It was heading west at about 14 mph, with maximum sustained winds of 45 mph.
A man died Saturday in Haiti while trying to cross a river in Leogane, south of Port-au-Prince, said Marie Alta Jean-Baptiste, head of Haiti's civil protection department. No further information was immediately available.
Rice fields in the Artibonite Valley, Haiti's most fertile region, were flooded, according to reports from Radio Ginen. And Fay's heavy winds destroyed banana crops in Arcahaie, north of the capital, although it is unclear how many acres were affected, Jean-Baptiste said.
Haiti has struggled to cope with a food crisis that sparked deadly riots in April.
The capital's airport reopened Saturday afternoon, but heavy rains were still expected in the south.
In neighboring Dominican Republic, a 34-year-old woman drowned when a family tried to cross a swollen river in a car, civil defense agency director Luis Luna Paulino said. The bodies of her missing 13-year-old niece and 5-year-old nephew were found Saturday afternoon, but her husband swam to safety.
A tropical storm warning has been issued for the Cayman Islands, and a tropical storm watch remains in effect for the Bahamas and Jamaica.
Cuba's government said hurricane watches were in effect for the provinces of Villa Clara, Cinefuegos, Matanzas, Camaguey, Ciego de Avila and Sancti Spiritus. Fay's path was expected to take it over the southern coast of the eastern Cuba late Saturday or Sunday and over the island's west near Havana on Sunday night or Monday.
Officials in Cuba's eastern province of Santiago met to discuss ordering tourists to evacuate low-lying coastal hotels and camp sites, the Communist Party newspaper Granma reported on its Web site. They also advised farmers to move cattle and other animals to higher ground.
There were no official details on how many tourists had been evacuated, but a receptionist at the Melia Santiago hotel, located in Cuba's second-largest city, said it was preparing to receive a "small number" of tourists evacuated from coastal areas. She said she was not authorized to be identified by name in the foreign media.
Forecasters said Fay could hit the U.S. as a Category 1 or 2 hurricane, with winds perhaps reaching more than 100 mph.