Securing Air Travel Act This bill addresses penalties for, and other matters pertaining to, carrying firearms beyond airport security checkpoints or on aircraft. Specifically, the bill establishes minimum civil monetary penalties for repeat or egregious violations of prohibitions against carrying unauthorized firearms at airports or on aircraft. It also makes individuals who violate the prohibitions ineligible for programs that expedite security screenings at airports. Additionally, the Transportation Security Administration (TSA) must display signage and carry out a campaign to inform travelers of the prohibitions and of the penalties for violating them. The bill also requires the TSA and the Government Accountability Office to report to Congress about the bill's implementation and related issues.