Betke received his Dr. med. in 1940 at the University of Berlin for his thesis titled, “Assoziationen bei genuiner und symptomatischer Epilepsie.” In 1953, he habilitated at the Albert-Ludwigs-University Freiburg with his work titled, “Human haemoglobin in the fetal and mature forms.” In 1962, he was appointed to the Chair of Pediatrics in Tübingen. Betke then succeeded Alfred Wiskott (1898-1978) at the Ludwig-Maximilians-University Munich. From 1967 until 1983, he held the position of Director and Professor of the Children’s Hospital, Children’s Clinic, and Pediatric Surgery at the Dr. von Haunerschen Children’s Hospital of LMU Munich.
Betke was highly esteemed in the field of pediatrics in Germany. In 1966 he became an honorary member of the Gynecology and Pediatrics section of the German Academy of Sciences Leopaldina. From 1968 until 1971, Betke was a member of the Science Council. In 1981, he was awarded the Bavarian Order for Science and Art. He was a member of both the Bavarian Academy of Sciences and Academy of Finland.
Betke has long been a member of the German Society of Pediatrics and has been an honorary member since 1991. In 1983, the German Society of Pediatrics awarded him the Otto Heubner Prize, a prize only awarded every three years, for his work with children. He has also received the Bavarian Order of Merit. Additionally, he is an honorary member of the University of Regensburg and has received honorary doctorates from the medical faculty of the Ruhr University in Bochum.