Yeah, everybody has their opinion on this one, and I'm no exception. What surprises me, though is that some people are so dogmatic about it.
Often the story first appears as a manga, and is then translated into an anime. Sometimes the anime comes first. Sometimes the source material is a light novel, or even some piece of heavy literature (Les Miserables comes to mind as a recent example). Maybe I'm more open-minded than other people about this because of taking a few classes in literary criticism in my college days. One thing that postmodernism was good for was opening peoples' eyes about what constitutes literature worthy of study. It's a core tenet of postmodernism that all texts (text here defined as anything that tells a story, including film, tv shows, etc) are created equal as objects of study. Like most things, they went too far with it, and so there's some utter crap now being seriously reviewed by academics. This is not entirely a bad thing - at least they're less likely to miss good stuff now.
If I haven't convinced you on this point, I'll remind you that Shakespeare's plays were populist universal entertainment performed (usually) in an open-air theatre on the wrong side of the river (that's like saying "the wrong side of the tracks," because there weren't tracks yet) in London. In a very real sense, plays like Love's Labors Lost are the direct ancestors of manga like Fruits Basket or Maison Ikkoku. Likewise most of the novels from the golden age of the prose novel were not regarded as serious literature in their day, either. Serious literature was epic poetry - stuff like The Faerie Queene, Paradise Lost, or The Idylls of the King (to choose a later example). - prose novels were trashy reading for the masses. It may seem strange to regard something like Moby Dick, Pride and Prejudice or Les Miserables as popular fiction, but that is what they were when initially published.
So with that said, the first thing you must do is imitate the academics (after all, they think about this kind of thing for a living) and accept that manga and anime are both forms of literature (pop lit, to be sure, but lit nonetheless) and that one is not inherently superior to another.
Now the next big step: allow for detatchment - the offspring need not be an exact copy of the parent. Frankly, I wish the Japanese did a little better with this one. Example: Stephen King wrote an OK supernatural horror book called The Shining. Stanley Kubrick made a very good psychological horror movie called The Shining, which was loosely based on the King novel.
If you're a hardcore Stephen King fan, you probably don't like the movie very much, because it's not especially faithful to the King story. It's a pretty easy case to make, though, that as a movie, the movie is a better piece of literature than the book is - the plotting is tighter, the characters are more vivid, and, frankly, King has always needed a cruel editor with a sharp knife where his novels are concerned - his best work is his short stories and novellas.
And so it goes with anything - the Karin anime has to be judged on its own merits as a show. Maybe it doesn't tell the same story that the manga does (in this case it definitely doesn't). That doesn't matter. What matters is how well it tells whatever story its attempting, and if the story is worth telling in the first place. And no, I don't particularly like the way the Karin anime is plotted for all that I'm quite fond of the manga.
That said, I think it's fair to say that if the manga came first, it's often better than the anime, simply because adding more cooks to the kitchen seldom improves the product. Or, to put it another way, the mangaka had a concept in mind when writing the manga. The anime producer (who often thinks of himself as an artist forced to deal with somebody else's art) seldom has the same concept in mind when plotting the anime. Two different (sometimes conflicting) views of who the characters are and what the story should be about. There also often seems to be a repurposing of content - Seinen manga are bowdlerized into all-ages anime entertainment (Chobits). Shounen manga sometimes get a fanservice and adult themes infusion in the anime to give them appeal to older viewers (Midori no Hibi).
On the other hand, there are a few situations (similar to the Stephen King example above) where the manga is too long, baroque and complex and the anime producer successfully distills it down into its essentials and makes it into a better story as a result. These are, however, in the minority.
The stories that seem to reliably survive the manga-anime translation the best seem to be character-driven talky mangas - Hataraki Man, Honey and Clover, Bokura ga Ita, Kamichu!, Mushi-shi, etc. Frequently, the action-intensive manga seem to get too frantic when translated to anime. Taken too far, however, you end up with talky anime, which isn't load of fun. Consider how hard Kyoto Anime had to work to make the Suzumiya Haruhi stories (which are fairly cerebral and very talky light novels) work as anime. Sometimes, the story is so obviously visual that it's hard to figure out how it could have been popular as a manga, but it is a shoo-in as a good anime. Binchou-tan is a recent example.