Let me first say that this is really two questions, the first, "Is writing art?" is simple enough to answer and I'd wager most(myself included) would say yes.
Then comes the second, "Does this make the quality of writing subjective?" is a bit trickier.
You see when I see people discuss drawing or music or other arts they all end up saying that while they like something it's more about personal preference, that the quality is at the end of the day subjective. And I wonder that if writing is an art then isn't the same true? Doesn't taste have as much an influence as actual writing skill? For instance, I know plenty of smart people who like Twilight, now I haven't read Twilight but that's because I am convinced the writing is crap. But is it really? Or is the problem that it's simply not the kind of story or writing that I enjoy? There are plenty of examples of this sort of thing, one that comes to mind is Dan Brown's stuff. I tried reading his books but I couldn't stand them at all, I felt like the writing was just the worst and yet I've heard plenty praise him as an author.
Personally I'm leaning towards quality of writing being subjective to at least the point of drawing. There's still an overarching "quality" to a given style but preference seems to be important too.
So anyone else got an opinion on this?