Even if you have the website open for anonymous access and anonymous users can access your documents in a document library, opening an office document will always require entering username and password. You can open the document, if you cancel the request for username and password three times. Still this can be annoying.
Untill now I was avoiding this by publishing PDF documents and making links to them. But you can change this. Let's say we're in a subsite /sub1 and have a Word document called mydocument.doc in a document library called Docs.
Linking to /sub1/Docs/mydocument.doc will request username and password from anyone opening the document. To avoid this, correct the hyperlink in your refference to:
is going to result in a simple download.
Now I can publish also Office documents for public.
UPDATE: Seems I was making the mistake of testing this authenticated. The _layouts/download.aspx requires authentication, so we're still stuck with the same problem. But it doesn't ask for username when opening word :)