When you run a MediaWiki installation behind basic authentication, VisualEditor reports this error:
Error contacting the Parsoid/RESTBase server (HTTP 401)
This happens because VE does not pass the basic auth credentials with the internal request to rest.php. A workaround is to edit .htaccess to allow requests from 127.0.0.1 to bypass the basic auth, but that's a bad idea on shared hosting because it means that other users can access your wiki contents. Not all shared hosting providers even support such allows in .htaccess.
My suggestion is to forward authentication headers to the internal request, or add an option for that, so that VisualEditor can work with basic auth without needing workarounds at the level of the web server.