Steve Swallow was born on October 4, 1940, in Fair Lawn, New Jersey. He studied piano and trumpet as a child, took up the double bass at 14, and began improvising while at prep school. At Yale he studied composition and played mostly with Dixieland bands. In 1960, at about age 20, he took a gig with the avant-garde pianist Paul Bley at a nearby college, went home, and left Yale to move to New York City and play jazz full time. Bley helped him land a job with the clarinetist Jimmy Giuffre, and it was through Bley that Swallow met Bley's then-wife, Carla Bley, who would become his closest musical partner.

The Giuffre trio with Bley and Swallow, formed in 1961, worked without a drummer and in open improvisation. It recorded Fusion and Thesis for Verve in 1961, played European concerts that year (tapes from Stuttgart, Bremen and Graz were issued decades later), and made Free Fall, released by Columbia in 1963. In the same years Swallow recorded three albums for Riverside with the composer George Russell (Ezz-thetics, with Eric Dolphy in the sextet, The Stratus Seekers and The Outer View, 1961 and 1962) and played on Paul Bley's Footloose (recorded in 1962 and 1963) and on Sheila Jordan's Blue Note debut Portrait of Sheila (1963). In 1963 he joined Art Farmer's quartet, which also featured the guitarist Jim Hall; he began writing music professionally in that group, and Hall recorded "Eiderdown", the first tune of his to be recorded. He also worked with Zoot Sims and Stan Getz, and he is on The Stan Getz Quartet in Paris, recorded live on November 13, 1966, with Gary Burton and Roy Haynes.

Swallow's long association with the vibraphonist Gary Burton began in the mid-1960s. He is on Burton's The Time Machine (1966) and on the quartet records that followed, among them Duster (1967), with the guitarist Larry Coryell and the drummer Roy Haynes, Lofty Fake Anagram (1967) and A Genuine Tong Funeral (1968), a suite of Carla Bley compositions. While working with Burton at a music trade show he tried a Gibson EB-2 electric bass; with Burton's agreement and Haynes's encouragement he began using it in the group, and by the early 1970s it was the only bass he played. He plays it with a copper pick, favors a five-string model with an added high C string, and uses instruments built by Harvey Citron. From 1974 to 1976 he taught at Berklee College of Music, where students assembling the first edition of The Real Book used several of his compositions; he later recorded an album called Real Book (1994) with Tom Harrell, Joe Lovano, the pianist Mulgrew Miller and Jack DeJohnette. His duo record with Burton, Hotel Hello, was recorded over two days in May 1974 and released by ECM in 1975.

In 1978 Swallow joined Carla Bley's band, and from the 1980s he was also her romantic partner. He is on the Bley albums Musique Mécanique (1979), Social Studies (1980), Heavy Heart (1984), Sextet (1987), Fleur Carnivore (1989) and many of the later big-band and chamber records. He and Bley run the Watt and XtraWatt labels, which issued his own records Carla (1987), Swallow (1991), Real Book, Deconstructed (with the trumpeter Ryan Kisor) and the live Always Pack Your Uniform on Top, recorded live at Ronnie Scott's Club in London and released in 2000. His ECM album Home (1980), built on poems by Robert Creeley, featured Sheila Jordan, the pianist Steve Kuhn, the saxophonist Dave Liebman, the synthesist Lyle Mays and the drummer Bob Moses. In the same years he played and recorded with John Scofield: Bar Talk (1980), Shinola (1982, a trio with the drummer Adam Nussbaum) and Out Like a Light (1983). He also produced a run of Scofield's studio albums, from Electric Outlet (1984) through Grace Under Pressure (1992).

From 1994 Swallow, Bley and the saxophonist Andy Sheppard made four albums as a trio: Songs with Legs (1994), Trios (2013), Andando el Tiempo (2016) and Life Goes On (2020), the last of them before Bley's death in October 2023. Alongside that he kept several other bands going: We3 with Nussbaum and Liebman (Three for All, 2006; Amazing, 2011), trios with the saxophonist Ohad Talmor and Nussbaum (Playing in Traffic, 2009; Singular Curves, 2014), the Swallow Quintet with Chris Cheek, Steve Cardenas and Jorge Rossy (Into the Woodwork, recorded in 2011 and released in 2013), and three albums with the keyboardist Jamie Saft and the drummer Bobby Previte, the second of which, Loneliness Road (2017), features the singer Iggy Pop. He rejoined Scofield for Country for Old Men (2016) and for Swallow Tales (ECM, 2020), a trio album with the drummer Bill Stewart built on Swallow's own tunes. In September 2024 he recorded Winter Songs with a sextet of Cheek, Cardenas, the trumpeter Michael Rodriguez, the pianist Gil Goldstein and Nussbaum; ECM released it on June 26, 2026. He has won the electric bass category in the annual DownBeat critics' and readers' polls consistently since the mid-1980s.

Swallow came up as a double bassist in the free-leaning groups of Jimmy Giuffre and Paul Bley, and he carried that melodic, exposed way of playing over to the electric bass: he favors the instrument's upper register and often approaches it as if it were a guitar. He has said that what first drew him to the electric bass was a clearer, more explicit voice, which the pick and the tone controls gave him, and that over the years he has looked for more ambiguity in his sound. His working vocabulary is post-bop and chamber jazz, with the harmonic openness he learned in the Giuffre trio. As a composer he has had his pieces recorded by Jim Hall, Bill Evans, Chick Corea, Stan Getz and Gary Burton. Carla Bley, who writes for him as a house composer, encouraged him to extend his range, and some of her parts require the high fifth string; he in turn often writes with her in mind, as on his 1987 album Carla. He was among the first jazz bassists to move entirely to the electric bass guitar, alongside Monk Montgomery and Bob Cranshaw.