Forecasters warned that Fay could reach hurricane force as it approached the Florida peninsula.
The storm's centre crossed Florida Key West with winds near 95km/h (60mph), flooding some roads.
Florida's Governor Charlie Crist warned that Fay threatened a "major disaster", but he insisted the state was prepared.
With a state of emergency in place, Mr Crist said some 500 national guard members had been deployed, with another 8,500 on standby.
The US National Hurricane Center said the storm still had the potential to increase strength by the time it hit the west coast of Florida, expected Tuesday morning local time.
At 0600GMT, the centre of Fay was located about 70km (45 miles) south of Naples on Florida's west coast.
The storm was moving at around 12km/h (seven mph) with maximum sustained winds of 95km/h (60mph), the NHC said.
 | HOW HURRICANES FORM Sea surface temperatures above 26.5C (79.7F) A pre-existing weather disturbance Moisture in the atmosphere Favourable conditions, such as light winds or weak wind shear |
Heavy rain was still hitting Cuba, which suffered some flooding but no major damage.
Fay's passage over Haiti and the Dominican Republic was more destructive, with more than a dozen deaths reported.
There was confusion over an accident in Haiti, where a truck carrying up to 80 passengers plunged into a swollen river on Sunday.
Officials and witnesses said more than half the passengers were missing, with many feared drowned.
However, UN peacekeepers stationed in Haiti who arrived at the scene told the Associated Press that two babies had died in the accident but that 41 passengers had escaped to safety.
A civil protection officer, who had earlier said some 30 people were feared dead, also told AP that three bodies had been found.
Fay is the sixth tropical storm of the 2008 Atlantic hurricane season.
Two of the tropical storms so far, Bertha and Dolly, have reached hurricane strength.