Ok, well if your site doesn't require visitors to log in, what you could do instead is use the Tag search page, which is a site page, therefore normally accessible to all visistors.
To promote it, provide a BIG link called something like "Search Courses by Keyword" at the top of your front page, normally accessible through <yoursite>/course/editsection.php?id=2. Get the link to go to <yoursite>/tag/search.php and see how it works.
You could also change the title of this page to something more suitable by getting an admin level account to use the language string customization feature documented here, at Site administration > Language > Language customization. Just be sure to search for the strings containing "Search tags"!
It is very straight-forward and simple to do via the GUI, and then your administrator just needs to clear the old language string caches by going to Site administration > Development > Purge all caches.
One thing to remember, this will show ALL tags on the site, so if it's possible, limit your tags to just course tagging, or at least disable the user contributed ones, else you could get some crazy or offensive words on this page.
If it absolutely has to be on the front page itself, you're probably looking at customisation work, to be done by a Moodle Partner.