Born in Elizabeth, New Jersey to a family of Ukrainian immigrants, Kasha studied chemical engineering at night at the Cooper Union in New York City for two years while working full-time during the days at the Merck & Co. research facility in New Jersey. He then received a full scholarship to the University of Michigan, where he completed a bachelor's degree in chemistry. He earned his Ph.D. in chemistry from University of California at Berkeley in 1945, working with renowned physical chemist G.N. Lewis. Following postdoctoral work with Robert Mulliken, he joined the chemistry department at Florida State University as a faculty member in 1951.