The web industry has matured to a point where requiring a related AS or BS will not limit the qualified candidates significantly.
Web designers should have a graphic design or visual communication degree. Their responsibility is to use color, layout, and design to communicate. If they lack the technical skills for the web, there are tools to help them - such as Adobe’s Dreamweaver, or even services that will convert their designs into HTML. For a pure designer, the technical skills to deliver the site should be secondary to the ability to make a nice page.
Back End engineers should have a CS degree. Whether they are developing custom applications or extending existing ones, the foundation knowledge offered in most CS programs is vital for the implementation of the requirements into cost-effective solution.
Server Admins must have a CS degree, and experience with the target platform. This is your last line of defense on the server from malicious people. It is also a place where you can make serious performance improvements.
Writers should have a degree in English, Marketing, or Communications. Don’t let the administrative assistant write the content unless they have a degree. The quality of the medium (in this case, the text) will affect its reception.
SEO probably doesn’t need any formal education. It is still such a new field that a well-read individual can do an adequate job, by researching independently. It is also possible that SEO is so difficult to actually perform that focussing on the site content is a more valuable investment than striving to make the content conform to the search engines.
There are definitely many excellent self-taught people in the industry, but they must constantly work to polish their skills. Newcomers into the field should have an educational background to enter the workforce ready to contribute and succeed.