CNN reached the migrants at a government shelter in San Ramon, a town about two hours northwest of San Jose, the capital. Most say they are desperate, even though the Costa Rican government has been feeding them and housing them in shelters.
Armando Quesada, one of about 300 at the shelter in San Ramon, said the group is running out of options. "Being smuggled by traffickers is not what we wanted, but we've been seriously considering that alternative since last night," Quesada said.
The largest group of Cubans has been staying at a shelter in La Cruz, in Guanacaste province, which borders Nicaragua.
Cuba has recently eased travel restrictions, allowing many in the island nation to travel for the first time in decades.
Most of the Cuban migrants now stranded in Costa Rica started their journey in Ecuador, a country that does not require visas for Cubans. From Ecuador the migrants traveled north by land through Colombia, Panama and Costa Rica.
The problem started when the first waves entered Nicaragua. The government sent police and soldiers to not only detain them but send them back to Costa Rica in buses. Now Nicaragua has sealed its borders, and the migrants say they don't want to stay in Costa Rica.
Costa Rican President Luis Guillermo Solís eased the migrants' worries by reiterating that his country will not send them back to Cuba in a message posted Wednesday on YouTube.
"Some of you are worried that the government of Costa Rica might negotiate a forced return of all of you to Cuba. That's not going to happen. I'm not going to do that," Solís said.
'A very sad situation'
The president also criticized Nicaragua's decision to block the Cubans.
"It has certainly been regrettable that that government of Nicaragua, in a move that is still incomprehensible to me, has denied free transit through its territory. This attitude, in my opinion, harms the spirit of integration and fraternity of Central America. It also constitutes for me a very sad situation in a Central America that I believed was more united."
For the most part, the Cubans say they're grateful to the Costa Rican government, but they still want to continue their journey north, according to Ivan Oliva, one of the migrants at the San Ramon shelter.
"We would like to thank Costa Rica for everything it has done for us, the help for children, the elderly and pregnant women; but we don't want to stay in Costa Rica. We want to go on toward the United States," Oliva said.
The government of Nicaragua accused Costa Rica of generating "a serious crisis" and of "violating treaties, borders and rights."
Costa Rica replied it's not the cause nor the origin of this situation and that its role is simply to protect these migrants as they travel through its territory.