United States - Mexico Border Health Education and Training Act of 1986 - Requires the Secretary of Health and Human Services, from amounts allotted to each State, to make grants to and enter into contracts with schools of medicine and osteopathy to establish and operate health education and training center (border center) programs along the border with Mexico (in Arizona, California, New Mexico, and Texas). Requires such border center programs to be cooperative programs between such schools and one or more private or public health education centers located along the border. Requires each border center to have an advisory board. Authorizes appropriations for FY 1987 through 1989.