Brian,
I'm assuming you will have more than one account that can do the download? If only one, you should be able to look at the stats in the metadata and see number of downloads, even broken down by date.
If more than one account, and you need a breakdown...maybe have different front ends (UI) to same download for each account? That's just thinking outloud.
I looked at a few of the utils including ArcGIS Online Assistant and (loging in to your enterprise account and viewing) Applications | ArcGIS for Developers but didn't see any break down.
Have you looked at any of the other admin tools?
Home · Esri/ago-admin-wiki Wiki · GitHub
Resources · Esri/ago-admin-wiki Wiki · GitHub
Otherwise, you may need to look at Google Analytics or similar tool. I have not used GA so can't offer any advice....but I guess you've already thought of that since I see you had a thread on that already
Just in case you haven't thought of all this already.
