Based on Petros answer, I used the two following curl command:
To get the list of all releases including their id and number of
download:
curl -i https://api.github.com/repos/:owner/:repo/releases -H "Accept: application/vnd.github.manifold-preview+json"
For example to list all the release for the OpenRefine project:
curl -i https://api.github.com/repos/openrefine/openrefine/releases -H "Accept: application/vnd.github.manifold-preview+json"
Then to get details on each release (you will need to run the first query to get the release id)
curl -i https://api.github.com/repos/:owner/:repo/releases/assets/:release_id -H "Accept: application/vnd.github.manifold-preview+json"
With the same example to list the details including download number for google-refine-2.5-r2407.zip
curl -i https://api.github.com/repos/openrefine/openrefine/releases/assets/6513 -H "Accept: application/vnd.github.manifold-preview+json"