Visa Security Expansion Act This bill requires the Department of Homeland Security (DHS) to increase the number of personnel in visa security units at certain diplomatic and consular posts. DHS shall increase the number of such personnel in at least two posts each year during a 10-year period. The bill establishes criteria for deciding which country's posts shall receive additional personnel, such as the number of nationals of that country who have been identified as a suspected terrorists and whether that country has been cooperative with U.S. counterterrorism efforts. In addition to existing duties, such personnel shall screen visa applications against U.S. criminal, national security, and terrorism databases. DHS shall establish a security fee for all nonimmigrant visa applications to offset the costs of operating and increasing the size of the visa security units.