Born in Berlin, and descended from Transylvanian Saxons, Baußnern was the son of a financial official. He grew up in both Transylvania and Budapest in the Austro-Hungarian Empire (in present-day Romania and Hungary respectively). Between 1882–1886 he was a student of Friedrich Kiel and Woldemar Bargiel at the Berlin Musical Academy (Berliner Musikhochschule). He then conducted various choirs; after 1909 he became director of the Großherzoglichen Musikschule in Weimar. From 1916 to 1923 he served as director of the Hoch Conservatory in Frankfurt am Main. In 1923, he became undersecretary of the Academy of Arts, Berlin. He died in Potsdam.