We are all painfully aware of the problem of people submitting copyrighted anime/manga images to deviantART, so I won't go into the details of the problem. Instead, I'm going to focus on why it's a problem and the solutions for it.

What is a copyright violation?
If you upload someone else's image into your gallery, then that is a violation of site policy. It doesn't matter if you filtered it, cropped it, made it into a wallpaper or sloppily altered the color in Paint, it's still a violation! Anime screencaps, manga scans and other images that you did not make yourself are against the rules!

Related FAQ:
FAQ #157: Can I use things created by other people in my submissions?
FAQ #8: What does deviantART consider to be Art Theft?
FAQ #304: Do you remove copies and trace-over art?
FAQ #257: What sort of permission do I need to use someone else's work?
FAQ #193: How do I prove that I had permission to use someone's work?
FAQ #94: Can I use the images I find on deviantART?

Why you shouldn't use copyrighted images:

1. It cheats the other artists. The rest of us work diligently to improve our art. If you upload images from a manga or anime, you are undermining everyone else and detracting from the real art at the site. When I search for "Naruto," I want to see fanart, not horribly-mutilated screenshots.

2. It cheats you. If you only upload other people's art (even if you modify it!) then you won't be able to improve as an artist. You don't want to be drawing stick figures at age 40, do you?

3. It cheats deviantART. DeviantART is a commercial art site. Taking someone else's art and uploading it here is theft.

4. It disrespects authority. When you signed up at deviantART, you committed to obey the site policy. Site policy prohibits using copyrighted images without permission. If you can't even speak Japanese, I sincerely doubt that Rumiko Takahashi gave you special permission to upload that oh-so-kawaii-squeal Sesshoumaru manga image.

5. It bugs the admins. If I had to wade through tons of screencapped violations every day, I wouldn't happy. Since most of the admins (with the possible exception of a self-proclaimed alien) are human, chances are that they find the concept annoying, too. Remember, kids- a happy admin is a less-likely-to-ban-you admin. If you upload copyrighted material over and over and over again, eventually their patience will wear thin- and you ARE violating site policy when you do steal images, after all.

6. It ruins the whole purpose of the site! DeviantART is an ART site to display YOUR work. It is not an image-sharing forum. With thousands of anime sites and free-image hosting sites on the internet, there is really no excuse for using deviantART as a Photobucket substitute.

7. It makes all anime artists look bad. Those of us who work don't like being stereotyped as art thieves- but most of the image-theft/copyright violations seem to involve anime images. The art community shouldn't be about anime-artists versus non-anime-artists, but when you portray us as being perverted, copyright-infringing tweens, it makes the anime fans who read and obey site policy look very bad. Keep in mind that your behavior (to outsiders, at least) reflects on the entire anime community! Don't make us look bad! (On a related note, most of the pedophilic images seem to turn up when one searches for "yaoi," which makes anime fans look even WORSE. If your mature "art" involves someone under 18, keep it on your hard drive. I don't want deviantART being sued into non-existence!)

Solutions:

I realize that you want to share your anime images anyway, so here some are ways to do it without violating site policy:

For hosting general anime images (screencaps, etc.)

Sign up at Photobucket. Photobucket is a free image-hosting site that offers free users up to 1 gigabyte of storage space. You can create folders within that album and tag images with keywords. For instance, you could have a "Naruto" folder, a "Bleach" folder, an "unknown series" folder, etc. If you want to share the images with your friends, you can link to them in your journal or comments without violating site policy. If you have a subscription, you can even directly embed them into your journal. Photobucket has long been a refuge for people hosting anime images. Please note, however, that you SHOULD NOT host images from deviantART at Photobucket. A printing capability the site recently added raises the possibility of art theft. Pornographic images are not permitted.

Use ImageShack. You can use ImageShack to host images without even signing up for an account, but if you sign up for a free account, it helps you to keep track of your images. Usage of ImageShack is similar to that of Photobucket- link to the images or embed them in subscriber journals. ImageShack does not currently support folders, but it allows you to tag images with keywords and there is a handy FireFox browser plugin which enables you to right-click any image and upload it to ImageShack. Pornographic images are not permitted.

Sign up with LiveJournal. LiveJournal is a free blogging community with many popular anime clubs and anime-related blogs. Ad-powered Plus accounts come with 1 gigabyte of free image storage, which can be organized by folder and keyword using an advanced system. The journal's layout is highly customizable with pre-made themes and supports custom-CSS. LiveJournal permits mature content (as long as it is not illegal,) but mature-content journals must be hidden behind lj-cut coding and marked as mature. The journals also integrate with Photobucket, if you have an account there.

For hosting wallpapers, avatars and scans:

LiveJournal has several communities dedicated to sharing these and offers 1 gigabyte of storage space for free Plus accounts.

MiniTokyo.net is a site dedicated to high-quality anime/manga scans, wallpapers and vectors. Ideally, you could upload your fanart to deviantART and your wallpapers to MiniTokyo.net. The MiniTokyo.net system for earning download points is simple and easy, with minimal effort.

Anime Paper is similar to MiniTokyo.net, but the image standards are much harder to meet and the download points are much harder to earn. Images accepted at MiniTokyo.net may be rejected at Anime Paper. However, there are several rare images that are available at Anime Paper, but not MiniTokyo.net.

There are numerous other free-image hosting sites and anime-themed gallery sites which offer free membership and use. If none of the more famous ones I have listed suit your needs, then search to find one that will. I have personally tried all of the sites I've listed in this article, and I have been pleased with all of them (even Anime Paper, despite its rigorous standards.)