Let me preface this by saying that I've not read the Twilight series. I've read enough about it from people who have read it to know that I'd probably prefer to keep it that way. And most of it is not the story line (although I have my issues with that), it's because of the writing. Yes, I've read enough descriptions of the writing to know that I'd prefer to avoid the writing (ones that gave examples from the books).
I'd prefer it if my children don't read Twilight though. And part of it is the story line. I'd prefer my children not think that the characters in the story are good role models. I'd prefer they don't think that it's okay to break and enter to watch someone sleep or to get depressed because someone breaks up with you (for your own good, in their opinion) and then start pulling stupid, suicidal stunts. And hopefully, even if they do read the books (because I won't stop them, even if I discourage it), I will have taught them enough to realize why they should look to other sources for good role models.
I'd prefer, honestly, that my children are inspired by someone like Hermoine Granger (with less rule breaking, preferably) or River Tam, or River Song for that matter. Someone who found love, but was strong even without that. Someone who had a fully developed personality. Someone who shows that doing the right thing is more important than some teenage relationship.
I want my children to read freely, from lots of sources. I plan on reading lots of books, lots of different types of books, to them when they are young. I hope to cultivate a lifetime love of reading. I don't want them to see books as my husband does; non-fiction preferred to fiction, movies preferred to books. Not because there is something wrong with non-fiction or movies, but because there are wonderful worlds out there in fiction.
I'd rather they keep an open mind.
I'd prefer they read and re-read Harry Potter, The Chronicles of Narnia, The Lord of the Rings, and similar books. I'd prefer that if they are interested in vampire fiction, they go for Dracula or Interview with a Vampire (even if I couldn't quite finish that book).
But I won't ban Twilight. I don't really plan on banning any books, if I can help it. I plan on encouraging them to read books that others have banned. Including Harry Potter. And Fahrenheit 451. And To Kill a Mockingbird. Because there are important things in these books. Some of those things are what got them banned in the first place.
But I want my children, especially my daughters, to have strong female role models. To know that they can be awesome and strong. To know that they can change their world.
And I'm not sure Bella Swan and sparkly Edward Cullen are the way to do that.