Border Business COVID-19 Rescue Act This bill requires the Small Business Administration to establish a loan program to assist border businesses that are directly impacted by the COVID-19 (i.e., coronavirus disease 2019) pandemic. For the purposes of this bill, the term border business refers to a business that (1) is eligible for an economic injury disaster loan, (2) has its principal office located in the contiguous United States and is located within 25 miles of the U.S. border, (3) has fewer than 50 employees and annual revenue of $1 million or less, and (4) demonstrates a reduction in revenue from specified prior periods. Recipients must use loan funds for purposes related to COVID-19 such as (1) providing sick leave to employees, (2) meeting increased costs arising from interrupted supply chains, or (3) paying for logistical expenses associated with border closures.