The storm formed one day before the official start of the season June 1, hitting land near the Mexican port city of Chetumal and Belize's Corozal city. It dumped rain as far south as Belize City and kicked up strong surf on the popular tourist island of Ambergis Caye.
Tropical storm warnings were issued for Belize and Mexico's Caribbean coastline.
In the Mexican state of Quintana Roo, which includes the popular resort of Cancun, ports were closed and all water sports were banned. Residents and tourists were encouraged to take precautions in coastal areas, said state Civil Protection Director Carlos Rodriguez Hoy.
Ports were also closed on the islands of Cozumel and Isla Mujeres and in Chetumal.
Authorities expected rains of up to a little more than 1 inch (30 millimeters) due to the passing remnants of Arthur, Rodriguez said.
In northern Belize, the National Emergency Management Organization expected about 4 inches (102 millimeters) of rain and warned of possible flooding around the Azul Hondo River.
Rain and rough seas ruined vacations for tourists in Ambergis Caye.
"I just came to lay in the sun and get a nice tan, but so far there hasn't been any sunshine," said Debbie Fountaineau, a police officer from Lake Charles, Louisiana, who arrived on the island Thursday.
The storm was projected to weaken as it crosses the Yucatan before moving out into the Gulf of Mexico as a tropical depression early Sunday.
There was chance it could strengthen back into a tropical storm before hitting Mexico again south of Veracruz on Wednesday, said Jamie Rhome, a meteorologist with the Hurricane Center. It was not expected to become a hurricane.
At 5 p.m. EDT (2100 GMT), the center of the storm was located about 75 miles (120 kilometers) northwest of Belize City, and about 195 miles (314 kilometers) southwest of Cozumel, Mexico. It was moving to the west-northwest near 7 mph (11 kph).
The storm was expected to stay well away from the U.S. Gulf Coast.