Alex is a murderer and rapist who likes…Beethoven. And he doesn't just like Beethoven; he reveres him. Alex is on a first-name basis with the composer, calling him "Ludwig van." And although Alex has no regard for women, when Dim makes fun of a woman humming Beethoven, Alex slaps him. Anyone who likes Beethoven is a friend of Alex.
The film uses Beethoven as shorthand for Alex's intelligence or sensitivity…you know, the sensitivity and intelligence that totally disappears when he's assaulting people. The fact that Alex has a total soft spot for everyone's favorite deaf composer hints at reserves of feeling within Alex's tough psychopath exoskeleton.
The motif of Beethoven is then compounded when Alex is made to despise the sound of Beethoven's music via the Ludovico technique. The fact that Alex starts screaming in pain when presented with images of people being battered and raped is one thing, but the fact that he's brainwashed to loathe the sound of some of the most beautiful music ever written is something very different—and this underscores the inhumanity of the Ludovico technique. If you erase the horrible parts of a person's character, A Clockwork Orange seems to say, you'll also risk erasing the good parts of a person's character. And that's bad news bears.
After all, as Alex says, "He did no harm to anyone. Beethoven just wrote music."
Ironically, there are two times when people specifically use Beethoven to harm Alex. The catlady attacks Alex with a bust of Beethoven. Weirdly, Alex makes no comment on this, but perhaps it is meant to foreshadow that latter event, when Frank uses Beethoven to drive Alex to suicide.
Basically, even though Alex raped and killed people, the public becomes sympathetic toward him when they find out that listening to Beethoven makes him sick. (Would they have been as sympathetic if Alex could no longer listen to Justin Bieber?)
The film doesn't ultimately state its opinion on whether or not Alex's love of Beethoven is really something that should make us feel warm 'n' fuzzy toward him. But it does clearly state that art has the power of allowing people to see the humanity in others…whether or not that humanity is actually there. The characters in A Clockwork Orange rally around Alex because of his aural love affair with Ludwig van, and we as viewers of A Clockwork Orange rally around Alex because he is himself an artistic creation—we kind of have to; he's our protagonist.
Bottom line: Kubrick and Burgess were geniuses, and we as humans are a sucker for anyone who likes the same music as we do.