Baeumker was born in Paderborn to a gymnasium teacher. He studied philosophy, theology, and philology in Paderborn and later at the University of Münster, from which he obtained a doctorate in 1877. From 1879 he was a gymnasium lecturer in Münster.
Through the efforts of Georg von Hertling he was appointed professor of philosophy at the University of Breslau in 1883, where he remained until 1900. In that year he moved to the University of Bonn, and in 1903 to the University of Strasbourg, where he filled the chair vacated by Wilhelm Windelband. In 1912 he moved to the University of Munich, filling the chair vacated by his friend von Hertling. He died in Munich in 1924.
Baeumker was known in particular for his studies in the history of medieval philosophy.
- Das Problem der Materie in der griechischen Philosophie, 1890
- The founder of the "Beiträge zur Geschichte der Philosophie des Mittelalters" (since 1891)
- Die europäische Philosophie des Mittelalters, 1909
- Roger Bacons Naturphilosophie, 1916
- Der Platonismus im Mittelalter, 1916
- Petrus von Hibernia der Jugendlehrer des Thomas von Aquino unde seine Disputation vor König Manfred, Munich, 1920.
- Friedrich Wilhelm Bautz (1975). "Clemens Baeumker". In Bautz, Friedrich Wilhelm. Biographisch-Bibliographisches Kirchenlexikon (BBKL) (in German) 1. Hamm: Bautz. col. 343. ISBN 3-88309-013-1.