Invention of this word is usually credited to the Czech novelist Milan Kundera, who described it like this in his book The Art Of The Novel in 1988: “To be without a feeling for art is no disaster. A person can live in peace without reading Proust or listening to Schubert. But the misomusist does not live in peace. He feels humiliated by the existence of something that is beyond him, and he hates it”.
So a misomusist is not a passive ignorer of culture, but an active opponent of it. Active opposition to culture has been a characteristic of totalitarian governments, summed up by a famous saying: “Whenever I hear the word culture, I release the safety-catch of my Browning!”. (Often attributed in a different form to Hermann Goering, it was actually written by the German dramatist Hanns Johst in 1933.)
Presumably Milan Kundera coined the word in Czech, from which it was carried over into the English translation. He took it from the Greek misos, hatred, and mousa, learning (the word is also the source of the name of the nine muses of Greek and Roman mythology who presided over the arts and sciences).
He cannot claim sole credit, however, for he was pre-empted by Sir Edward Dering more than three centuries ago, in his Collection of Speeches in Matters of Religion of 1642: “Our better cause hath gained by this light: which doth convince our Miso-musists”. That was a once-off invention that was never taken up by others. Even after Kundera’s reinvention, it can hardly be called common.