In a publishing site even if anonymous users have been granted permission on all the level i.e in Central admin, in Site and if permissions are broken then in List\ Libraries but still they get prompted to enter username\ password to view the list\ list items.
To fix this issue deactivate a hidden feature by name ViewFormPagesLockDown, this feature when active denies access to anonymous users to all pages of type "application pages" by prompting authenication.
Below is the command to deactivate it, after deactivating it turn off\ on anonymous authentication on all the 3 level i.e
in CA for the webapplication,in Site collection and
finally in any lists that you have broken permissions.
stsadm -o deactivatefeature -url http://yourserver -filename ViewFormPagesLockDown\feature.xml
Below is a MS link to configure anonymous access in sharepoint.