Bashkirov was born in Tbilisi, Georgia. He became an Honored Artist of the RSFSR in 1968 and a People's Artist of the RSFSR in 1990. He is a recording artist with the Swiss classical record label Claves, with which he has recorded works of Johann Sebastian Bach and Beethoven. As a pedagogue at the Moscow Conservatory and Queen Sofia College of Music in Madrid, he raised many internationally renowned artists such as Arcadi Volodos, Dmitri Alexeev, Dang Thai Son, Nikolai Demidenko, Elena Bashkirova, Kirill Gerstein, Denis Kozhukhin, Eldar Nebolsin, Vestards Šimkus, David Kadouch, Jong Hwa Park, Claudio Martinez Mehner, Bruno Vlahek, Plamena Mangova, Stanislav Ioudenitch and many others. Bashkirov has a daughter, Elena Bashkirova, who is married to Daniel Barenboim.