Kasha was named the Robert O. Lawton Distinguished University Research Professor at Florida State University in 1962, which is the university's highest honor. He was elected as a Fellow of the National Academy of Sciences in 1971, the first Floridian to be so honored. He was also a Fellow of the American Academy of Arts & Sciences (1963), as well as the International Academy of Quantum Molecular Science. He was an elected member of the Ukrainian SSR Academy of Sciences and the Brazilian Academy of Sciences, and received the Porter medal from the European Photochemistry Association (1990).