Kleene served as president of the Association for Symbolic Logic, 1956–58, and of the International Union of History and Philosophy of Science, 1961. In 1990, he was awarded the National Medal of Science.
Kleene and his wife Nancy Elliott had four children. He had a lifelong devotion to the family farm in Maine. An avid mountain climber, he had a strong interest in nature and the environment, and was active in many conservation causes.