A state of emergency has been declared in the Central American nations, where more than 20 people are missing.
It has caused heavy rains, landslides and floods which are blocking roads, destroying bridges and damaging houses.
In Costa Rica, nearly 400,000 people are without running water and thousands are sleeping in shelters.
At least six people have died in the storm there, while another 11 were killed when it moved north and reached Nicaragua.
Three people have been killed in Honduras and several are reported missing.
In Costa Rica, all train journeys were suspended and dozens of flights cancelled on Thursday.
More than a dozen national parks popular with tourists have been closed as a precaution.
The storm also caused extensive damage to infrastructure in Nicaragua.
Forecasters say Nate will gain strength and become a category 1 hurricane before it makes landfall on the southern coast of the United States on Sunday.
Oil companies operating in the Gulf of Mexico says they are evacuating staff from platforms which lie along the predicted path of the storm.
Photo copyright: REUTERS (Caption: Heavy rain has swollen rivers and affected towns across Costa Rica)