This extension supports DAV quotas, but the feature is not provided by Apache's mod_dav by default. This may be different for other servers. For more information on webdav, see webdav.org
To configure Apache's mod_dav to require authentication for writes but allow public access, you may use this configuration. GET and PROPFIND are still allowed.
<Limit PUT POST DELETE PROPPATCH MKCOL COPY MOVE LOCK UNLOCK>
# No directory listing with GET method
# No directory listing with webdav PROPFIND method
RequestHeader set Depth 0