Botschinsky's father played bassoon professionally. He began on trumpet at age 11 and at 14 began studying 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. In 1963-64 he studied at the Manhattan School of Music. From 1964 he played with the Danish Radio Jazz Group and also played with the Danish Radio Big Band from the late 1960s.
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. The conductor for the 1978 edition after Denmark's absence at the contest was not, as some have mistaken, Botschinsky but rather Danish musician Helmer Olesen.
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. Among those he has played with as a sideman are First Brass, Sahib Shihab, Bjarne Rostvold, and George Gruntz.
- Jazz Quintet 60 (1963)
- Duologue (1987)
- The Night (1988)
- Jazzpar 95 (1995)
- Bench (2006)