The central American country has been gripped by a year of political unrest since months of street protests erupted last April against President Daniel Ortega's government, leaving more than 325 people dead.
People "have resorted to irregular crossings to avoid detection, often walking for hours through difficult terrain, exposed to heat, humidity and the risk of malaria," UNHCR said in a statement.
Those seeking asylum in Costa Rica include "students, former public officials, opposition figures, journalists, doctors, human rights defenders and farmers," the agency added.
Government talks with the opposition aimed at easing the crisis broke up last week without agreement on political reform in the crisis-torn country.
Ortega's leftist government has said it remains committed to breaking the impasse through dialogue.
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