Kenneth Sidney Drew was born in New York City on August 28, 1928. He began piano lessons at the age of five and attended the High School of Music & Art in Manhattan. His first recording was a Blue Note session with trumpeter Howard McGhee's All Stars on January 23, 1950, in a sextet that also held J.J. Johnson, the tenor saxophonist Brew Moore, bassist Curly Russell, and Max Roach. Over the next three years he worked and recorded with several of the central figures of the music: a February 1950 quartet date with Sonny Stitt and Art Blakey, a Chicago club recording that April with Lester Young, a December 1950 Birdland quintet recording led by Charlie Parker, Birdland broadcasts in 1951 with a Miles Davis sextet that included Sonny Rollins, a December 1951 quartet date under Rollins's name, and the quartet of the clarinetist Buddy DeFranco in 1952 and 1953. He also played in bands led by Coleman Hawkins.

His own first album came in 1953: a trio session for Blue Note recorded on April 16 at Rudy Van Gelder's Hackensack studio with Russell and Blakey, released as New Faces - New Sounds. Trio dates for Norgran followed in 1953 and 1954 with the bassist Eugene Wright and a different drummer on each session. Drew then spent a stretch in California, leading a trio for a time and recording with Zoot Sims (July 1954), with the Clifford Brown and Roach septet on the Best Coast Jazz date (August 11, 1954), with Dexter Gordon (September 1955), and on his own quartet album Talkin' & Walkin' for Jazz:West (December 1955), with the altoist Joe Maini and bassist Leroy Vinnegar. On March 1 or 2, 1956, he was the pianist in the Jazz:West quartet led by bassist Paul Chambers, with John Coltrane on tenor saxophone and Philly Joe Jones on drums.

Back in New York he became a regular in the studios of the later 1950s. For Riverside he made the Kenny Drew Trio album with Chambers and Philly Joe Jones (September 1956), a Jerome Kern duo album with the bassist Wilbur Ware (with Harry Warren and Harold Arlen duo albums for Judson from the same February 1957 sessions), the quintet and quartet album This Is New with Donald Byrd and Hank Mobley (March and April 1957), and the trio album Pal Joey (October 1957). On September 15, 1957, he was the pianist on Coltrane's Blue Note album Blue Train, with Lee Morgan, Curtis Fuller, Chambers, and Philly Joe Jones. Sideman work through 1958 included Johnny Griffin's Riverside sextet date (February), Chet Baker's Riverside quartet album It Could Happen to You (August), and a pair of Birdland sets with Buddy Rich (November); Wikipedia also lists work with Dinah Washington. In 1960 and 1961 he was a Blue Note regular: with Jackie McLean and the tenor saxophonist Tina Brooks, with Kenny Dorham on Whistle Stop, with Grant Green on Sunday Mornin', with Gordon on Dexter Calling..., and as leader on Undercurrent, recorded on December 11, 1960, with Freddie Hubbard, Mobley, Sam Jones, and Louis Hayes, the last of his Blue Note leader dates.

Drew moved to Paris in 1961 and to Copenhagen in 1964, and he lived in Denmark for the rest of his life. In Paris he recorded with the guitarist René Thomas (March 1963), played on Griffin's quartet recordings from the Warsaw Jazz Jamboree in October 1963, recorded in a quartet led by the drummer Daniel Humair with Eric Dolphy (May 28, 1964), and was the pianist on Gordon's One Flight Up, recorded in Paris on June 2, 1964, with Byrd, the Danish bassist Niels-Henning Ørsted Pedersen, and Art Taylor. In Copenhagen he, Pedersen, and the drummer Alex Riel became the regular rhythm section at the Jazzhus Montmartre, backing the American players who came through the club. Recordings from the club and from Copenhagen studios place him with Ben Webster (from January 1965), the violinist Stuff Smith (March 1965), Gordon (a run of dates from June 1965 into the 1970s), Brew Moore (February 1966), Hawkins (February 1968), Mobley (March 1968), and Rollins (September 1968). According to AllMusic, he was also a co-owner of the Matrix label in Copenhagen. In 1971 he and Gordon wrote and played the score for Ole Ege's film Pornografi.

The Copenhagen label SteepleChase, founded in 1972 by Nils Winther, who began by recording concerts at the Montmartre, became his main outlet for leader dates. He made Duo with Pedersen on April 2, 1973, followed by the solo album Everything I Love, Duo 2, and Duo Live in Concert, and in May 1974 the trio albums Dark Beauty and If You Could See Me Now with Pedersen and drummer Albert "Tootie" Heath. Later SteepleChase albums paired him with guitarist Philip Catherine and Pedersen (Morning, 1975, and In Concert) and, for Lite Flite (February 1977), with Thad Jones, tenor saxophonist Bob Berg, bassist George Mraz, and drummer Jimmy Cobb. As a sideman for the label he played on Gordon's The Apartment, Griffin's Blues for Harvey, and McLean's A Ghetto Lullaby and The Meeting. Wikipedia credits Pedersen and Drew with more than fifty albums together. A trip to Los Angeles produced Home Is Where the Soul Is and For Sure! for Xanadu (October 1978, with Charles McPherson on the second). In the early 1980s he made albums for Soul Note in Milan, where in February 1983 he also played on the saxophonist Claudio Fasoli's Lido and the singer Tiziana Ghiglioni's Sounds of Love, and in May 1985 on Lilian Terry's album with Dizzy Gillespie; he also made a long series of trio albums for the Japanese label Baystate with Pedersen and the American drummer Ed Thigpen; from 1988 the Alfa label recorded him with Pedersen and drummer Alvin Queen; and in March 1990 he joined Hank Jones, bassist Mads Vinding, and Billy Hart for the two-piano album New York Stories.

Drew passed away in Copenhagen on August 4, 1993, at 64, and was buried at Assistens Cemetery in the Nørrebro district. His final recording, a live trio set with Pedersen and Queen, was made at the Blue Note club in Osaka in October 1992. A street in southern Copenhagen, Kenny Drews Vej, is named after him. His son, Kenny Drew Jr. (1958 to 2014), was also a jazz pianist, with a separate career and a separate discography.

Drew played in the bebop tradition: AllMusic's Scott Yanow described him as a bop-based pianist, and The Biographical Encyclopedia of Jazz called his playing precise, a mix of bebop-derived melodic improvisation with block chords and refreshingly subtle harmonizations. His 1950s records, from the Blue Note trio date to the Riverside albums with Paul Chambers and Wilbur Ware, show a melodic hard-bop pianist who liked to build whole albums around one songwriter, as in the Jerome Kern, Harry Warren, Harold Arlen, and Rodgers and Hart sets. After the move to Denmark his work widened: he became one of the most recorded accompanists in Copenhagen, the dependable pianist behind Dexter Gordon, Ben Webster, Johnny Griffin, and Jackie McLean on club and studio dates, and he built a long duo and trio partnership with the bassist Niels-Henning Ørsted Pedersen. Drew himself said that living in Copenhagen put him in more varied musical settings than he would have met had he stayed in New York with a fixed group of players. AllMusic called him somewhat underrated because his decision to stay in Copenhagen kept him out of the American spotlight, though he built a wide following across Europe.