His background is in classical music. His father played bassoon professionally. He began on trumpet at age 11. At 14, he studied at the Royal Danish Conservatorium. In 1956 he joined Ib Glindemann's big band, where he remained through 1959; he also played with visiting American musicians around this time, such as Oscar Pettiford, Stan Getz, Dexter Gordon, Lee Konitz, Ben Webster, and Kenny Dorham. From 1963–64 he studied at the Manhattan School of Music. From 1964 he played with the Danish Radio Big Band.
While with Danmarks Radio, he worked as a conductor and arranger. He was the conductor for several editions of Dansk Melodi Grand Prix and conducted all of Denmark's entries in the Eurovision Song Contest from 1979-1983.
In 1985 Botschinsky relocated to Hamburg, Germany, where he worked with Peter Herbolzheimer and the European Trumpet Summit in addition to his own ensembles. He founded his own label, M.A. Music, with his sister Jette Botschinsky and Marion Kaempfert in 1987, though he also recorded for Storyville, Stunt, and Telefunken. He worked as a sideman with Kenny Dorham, George Gruntz, Nils Lindberg, Bjarne Rostvold, and Sahib Shihab.
- Duologue (M-A, 1987)
- The Night (M-A, 1988)
- Jazzpar 95 (Storyville, 1995)
- The Bench (Pony Canyon, 1996)
- First Brass (M-A, 2013)
- I've Got Another Rhythm (M-A, 2013)
- Last Summer (M-A, 2013)
with Oscar Pettiford
- 1960 First Bass
- 1962 Last Recordings by the Late, Great Bassist
- 1989 Montmartre Blues
with Ben Webster
- 1970 Masters of Jazz, Vol. 5
- 1970 No Fool, No Fun
- 1970 Plays Ballads
- 1989 Plays Duke Ellington
- 2006 Storyville Ben Webster
- 2007 Dig Ben!
- 2009 100 Years
with Dexter Gordon
- 1975 More Than You Know
- 1976 Strings and Things
- Yanow, Scott. "Allan Botschinsky | Biography & History | AllMusic". AllMusic. Retrieved 23 October 2016.
- "Allan Botschinsky | Album Discography | AllMusic". AllMusic. Retrieved 23 October 2016.