It depends a lot on where you are-- but-- as long as you're willing to put up with ads, the only legal way that I know of to get this kind of content in N. America is.... YouTube. Obviously-- that comes with some grey area, too, and the copyright issues mean that anything that appears may be transient (a.k.a. taken down without warning), but the alternative, as you've said, is to torrent.
MOST of DD's college friends see little problem with torrent as a means of obtaining content that isn't available other ways, and not a few of them justify high costs even as a good enough reason to do it-- but she (and we) are pretty uptight about it. Most ISP's have pretty harsh responses to suspected torrent users when it rises beyond "occasional" as well-- and they WILL tag you for it if your usage is high.
So all of that to note that I'd object, as well. It is possible to save YouTube clips, though it isn't necessarily easy to save them in HD, and obviously it requires a ton of external storage space- but an external drive might be an easy solution to that.
The other thing that I'd caution you about is that if your child is young-- heck, even up to mid-teens, I'd say-- you should probably be CAREFULLY vetting Anime content. Some of it is fine, but there is a subculture of very, er, unwholesome content there, and even the sort-of-okay-ish Anime content often portrays relationships, female characters, and other stuff in ways that are downright distressing and toxic. Hentai isn't far behind-- and that IS, unfortunately, the underpinnings of much Anime culture.
I know that you didn't ask-- but Anime often turns into a gateway to pretty sick porn, and parents don't see it coming because they don't understand that link. Asynchronous kids are obviously at even greater risk than most children there because of their mix of sophistication/savvy and naivete borne of a lack of life experience.
ETA: The other thing that we've had success with (not for Anime, but other video content from overseas) is cultivating a sort of, er.. cultural exchange. We have friends all over the world-- and those that I've known for years, have exchanged holiday cards with, that kind of thing? They are often more than willing to burn to a DVD and send them to us.
Obviously-- that violates (probably) the number one rule of internet safety on some level-- so it really needs to directly involve parents. But if you have friends in Japan, they might be willing to send material to you in exchange for something else that you can send to them. It seems more legal than torrent, anyway, and it's the only other way that I know of to get some of the more obscure media that is out there in full-length versions. (This used to be our only option for Doctor Who Christmas Specials, in fact.)
HTH.
