Is there a way within GitHub to see at the very least the overall number of users who have either seen or downloaded a repository?
GitHub API returns downloads count of binary file releases. I developed a little script to easly get downloads count by command line.
As already stated, you can get information about your Releases via the API. However, the
download_count value does not track users, just downloads.
For those using WordPress, I developed this plugin: GitHub Release Downloads. It allows you to get the download count, links and more information for releases of GitHub repositories.
To address the original question, the shortcode
[grd_count user="User" repo="MyRepo"] will return the number of downloads for a repository. This number corresponds to the sum of all download count values of all releases for one GitHub repository.