Border Health Security Act of 2014 - Amends the United States-Mexico Border Health Commission Act to revise the duties of the United States-Mexico Border Health Commission to include: (1) cooperating with the Canada-United States Pan Border Public Health Preparedness Council, and (2) serving as an independent and objective body to both recommend and implement initiatives that solve border health issues. Authorizes members of the Commission and the Council to provide advice or recommendations to the Secretary of Health and Human Services (HHS), Congress, or any Member of Congress concerning issues that are considered by the Commission or Council.

Requires the Secretary to award grants: (1) to eligible entities to improve the health of individuals residing in the U.S.-Mexico and U.S.-Canada border areas, and (2) for Early Warning Infectious Disease Surveillance to eligible entities for infection disease surveillance activities in such areas.

Requires the Commission and the Council to each: (1) prepare a binational strategic plan to guide the operations of the Commission and the Council, and (2) develop and approve an operational work plan and budget based on the strategic plan.

Requires the Comptroller General (GAO) to conduct an evaluation of Commission and Counsel activities.

Authorizes the Assistant Secretary for Preparedness and Response to coordinate with the Secretary of Homeland Security (DHS) in establishing a health alert system that: (1) alerts clinicians and public health officials of emerging disease clusters and syndromes along the U.S.-Mexico and U.S.-Canada border areas; and (2) warns of health threats, extreme weather conditions, disasters of mass scale, bioterrorism, and other emerging threats along those border areas.