Hensel was born in Königsberg, East Prussia (today Kaliningrad, Russia), the son of Julia (née von Adelson) and landowner and entrepreneur Sebastian Hensel. He was the brother of philosopher Paul Hensel. Kurt and Paul's paternal grandparents were painter Wilhelm Hensel and composer Fanny Mendelssohn. Fanny was the sister of Felix Mendelssohn Bartholdy, daughter of Abraham Mendelssohn Bartholdy, and granddaughter of philosopher Moses Mendelssohn, and entrepreneur Daniel Itzig. Both of Hensel's grandmothers and his mother were from Jewish families that had converted to Christianity.
Later in his life Hensel was a professor at the University of Marburg until 1930. He was also an editor of the mathematical Crelle's Journal. He edited the five-volume collected works of Leopold Kronecker.