I've shared a file on Google Drive publicly.
Can I see how many times the file is downloaded?
Many storage clouds have this possibility, I did not find this option in Google Drive. And that's missing me...
Google Drive hasn't a download counter. Send your feedback to Google by using Google Feedback.
The alternatives depend on how you share the links to your files but in any case once the user has accessed the file throw a link generated by the Google Drive web app, it will be shown on their Google Drive Shared with me and they will be able to add the file to My Unit / Private Workspace and/or save the direct link somewhere like the web browser bookmarks. On these cases it will not possible to track the downloads.
Another alternative is to create a web application that downloads the file, this way a temporary link will be used to download the file and your web application could do the count. This could be achieved by using Google Apps Script and/or Google Drive API and Google Picker.
Google Drive does not provide such a feature, but there are luckily several workarounds in a game how to get those numbers (unfortunately not retrospectively). For example, if you get yourself an account at https://bitly.com/ and wrap your download link into short (even custom) bit.ly URL and check the stats.