In 1964, at the Newport Folk Festival, Cajun guitarist Dewey Balfa exposed the world to music that was all but unknown outside of Louisiana. Today, Dewey’s daughter Christine and her band, Balfa Toujours, nurture new branches on the venerable family tree.

If someone would have told me that we would have five CDs and toured full time, I would have thought they were crazy." The voice is Christine Balfa’s and the band...