So there have certainly been isolated incidents of 12-tone music working its way into "pop" music (Zappa comes to mind, as does the use of 12-tone music as a topic in film scores), and some 12-tone music (Berg's Violin Concerto and Dallapiccolas Quaderno musicalae) is pretty listener-oriented. But no, I wouldn't describe Schoenberg's style as ever becoming "popular," at least in the sense that, say, rock and roll or punk rock became "popular."

But of course, two things should be kept in kind. 1.) Schoenberg was at his core a Romantic composer. This entials things like seeing musical history as something that does progress in the first place, and, especially early in his career, developing atonal approach precisely because he wanted pure feeling (ie, not rules grounded in tonal syntax) to be the only reason why one musical thing follows another (the 12 tone system is a response to some of the limitations of that approach). And 2), there is no way that Schoenberg could have imagined the booming recording industry that developed in the 1930s and beyond. The way people consumed music and our very conception of what it means to be popular were soon to change very radically. Pretty soon, to be popular was to have airplay, to be promoted by labels and DJs, to bombard listeners of radio stations and go platinum in your record sales. So no, Schoenberg's style did not become popular, but that's partly because the very distinctions between "popular music" and "classical music" were gradually becoming entrenched, and Schoenberg clearly fell on the latter side.

Now, Schoenberg's serialist techniques did have a big impact on a new generation of modernist composers like Milton Babbitt, and many other 20th century composers dabbled in the system, often adapting it radically in their own idiosyncratic ways (ie, Stravinsky). So in that sense, Schoenberg was an extremely important figure in the history of 20th century art music. It's just that when he was composing, he still bought into this romantic vision (evident in, say, ETA Hoffman's reviews of Beethoven Symphonies) that there was an ideal "endgame" of sorts for music, one in which audiences everywhere would "rise up" to appreciate the most challenging of artworks. That it was the "job" of a good composer to push boundaries, thereby "advancing" music and humanity as a result, who would eventually come to the "correct" view that what they were doing was worthwhile. This vision was pretty much destroyed by the segregation of pop and art music (though it was always a pipe dream in the first place). Composers and philosophers like Babbitt and Adorno eventually realized that what they were doing was simply never going to become "popular," though their reactions to this realization were varied and multifaceted.

In short, Schornberg was important because he represented one of the last great romantic efforts in music, as well as bequeathing essential aesthetic and technical means of music making to an emerging modernist musical school. In many ways, the fact that this music never caught on is evidence not only of how revolutionary Schoenberg was, but also for a shift in how culturally valuable it was to be revolutionary, to push boundaries in the first place.