The center said the storm was expected to stay out in the Pacific while moving northwestward along the Baja coast and continue weakening through the night, likely becoming a tropical depression by Friday evening. Still, it was raking the shore with powerful winds and heavy rains, creating the potential for dangerous flooding.
High surf already claimed two lives in the area. Police in Cabo San Lucas said a 15-year-girl was trapped by a large wave and an adult tried to save her Tuesday. Both died.
The hurricane center said Genevieve's maximum sustained winds had dropped to 65 mph (100 kph) Thursday night, and it was centered about 70 miles (110 kilometers) southwest of Cabo San Lazaro. It was moving to the northwest at 10 mph (17 kph).
The storm knocked out power and phone service to a large part of the Los Cabos area, flooded streets in poor neighborhoods and toppled palms in the tourist zone. Los Cabos Mayor Armida Castro said more than 800 people had gone to shelters in Cabo San Lucas and 250 sought refuge in San Jose del Cabo, where distancing measures were in place due to COVID-19.
Baja California Sur state officials said 15,000 foreign tourists were in the state, most in the Los Cabos region, which earlier had almost been emptied of visitors by pandemic restrictions.
Meanwhile, two new tropical depressions formed Thursday in the Atlantic Basin, and tropical storm watches were posted for several Caribbean islands and parts of Honduras.
The hurricane center said Tropical Depression 13 was likely to become a tropical storm Friday and then skirt the Leeward Islands, Puerto Rico, the Dominican Republic and Cuba. The early, still uncertain track showed it potentially being near Florida by Monday as a hurricane.
Late Thursday, it was centered about 445 miles (715 kilometers) east of the northern Leeward Islands with maximum sustained winds of 35 mph (55 kph) and it was headed briskly to the west-northwest at 22 mph (35 kph).
Tropical Depression 14 was forecast to graze the Atlantic coast of Honduras, then curve northward to Mexico's Yucatan Peninsula possibly at hurricane strength and then potentially head for the Texas or Louisiana coast, again possibly strengthening into a hurricane. It is expected to become a tropical storm on Friday.
Late Thursday, it was centered about 65 miles (110 kilometers) east of Cabo Gracias a Dios on the Honduras-Nicaragua border, with maximum sustained winds of 35 mph (55 kph). It was headed west-northwest at 14 mph (22 kph).