Pick your battles. If a younger child wants to watch anime, he will find a way even if you say "no." Try to find a compromise by using several "anime-style" titles from television stations such as the Cartoon Network. Also remember that although most anime is intended for older teenagers and adults, many television stations heavily edit anime content for younger audience viewing.
Start with Disney or Hayao Miyazaki. Although most anime contains mature life themes about death, love or warfare, most of the Disney anime-styled titles and Hayao Miyazaki titles leave out gratuitous sex or other mature content. These titles also contain valuable life lessons or themes about growing up or respecting others and explore such areas as gender, race, age and saving the environment.
Watch anime on the Cartoon Network, online or on other channels that list anime as a part of their line-up, and create a two column list for your age group of appropriate and inappropriate titles. You can not learn whether anime is age appropriate simply by reading a ratings sticker or a brief description on the back of a DVD box. Also, many heavily edited anime shows still contain unedited social themes or interactions such as the displaying of a young character's underwear, over-endowed physical "attributes" or the use of violence or harsh language by a character in a young age group.
Learn a new form of communication or maybe two. Japanese animation contains two forms of language that you may not be familiar with: emotional iconography and anime specific body language or gesturing. Emotional iconography is the depiction of a character's emotion with certain artistic markings such as sweat drops, tear lines, eye changes and super-deformation or the removal of all "cuteness." Anime character body language is also used extensively to depict emotion.
Add an additional language to your vocabulary by learning a few Japanese words used to describe anime. Not only will you be able to communicate better with your child or children, but you can also determine if a title is appropriate to a particular age group based on online descriptions such as "shonen-ai" (story about love between two young men) or "mecha" (story about "real" and super robots).
Network with other parents, teachers or school administrators in your area or online to find additional titles.