Securing American Families from Exploitation at the Border Act or the SAFE Border Act This bill directs the Federal Emergency Management Agency (FEMA) to establish a temporary grant program to reimburse landowners for loss or damage to property caused by an individual illegally crossing the southern border or to improve the security of such property. Specifically, FEMA may provide grants to landowners who have experienced (or are at risk of experiencing) property theft or damage in areas within 10 miles of the U.S.-Mexico border.