Born to Karl Klöpfer and his wife Karoline, née Hörsch, Eugen attended the Realschule (secondary school) in Heilbronn.
He subsequently attended the Lateinschule (Latin School) in Lauffen and then the Karlsgymnasium in Heilbronn.
Although he started an apprenticeship with a lumber business in Munich, Klöpfer soon discovered that his passion was the theater.
In 1940, he played the role of Landschaftskonsulenten Sturm in Veit Harlan's anti-Semitic Nazi propaganda film Jud Süß.
In August 1944, towards the end of the Second World War, Klöpfer was added by Adolf Hitler to the Gottbegnadeten-Liste, a list of important German artists, which exempted him from military service, including service on the home front.