Born in St. James Parish, Louisiana, Braud came to New Orleans in his early teens. He was playing violin and string bass and leading a trio in venues in the Storyville District before 1910. He moved to Chicago in 1917. In 1923 he went to London with the Plantation Orchestra, in which he doubled on bass and trombone. Next he moved to New York City where he played with Wilber Sweatman's band before joining Duke Ellington. His vigorous melodic bass playing, alternately plucking, slapping, and bowing, was an important feature of the early Ellington Orchestra sound in the 1920s and 1930s. Braud's playing on Ellington's regular radio broadcasts and recordings helped popularize the slap style of string bass playing, as well as encouraging many dance bands of the time to switch from using a tuba to a string bass. (Like many of his contemporary New Orleans bassists, Braud doubled on tuba, and he recorded on that instrument on some sides with Ellington.)
In 1936 Braud co-managed a short lived Harlem club with Jimmie Noone, then formed the group The Spirits of Rhythm. He played with other New York bands including those of Kaiser Marshall, Hot Lips Page, and Sidney Bechet, and returned for a while to Ellington in 1944. In 1956 he joined the Kid Ory Band with whom he stayed for years.
He died in Los Angeles, California.