I'm not aware of a out of the box solution for this. But it shouldn't be that hard to do. I would do the following:
1) Make sure all the files you want to track are copied/uploaded to a special subfolder in the assets directory (let's name it assets/filefolder. You could even protect that folder with a password, to deny direct download of files.
2) Create a controller that finds a file, writes a log entry to the statistics database and passes the file data to the user for download. The comments on the following page show how to output file-contents to the user: http://php.net/manual/en/function.fread.php
3) Create a custom rule for your controller (http://doc.silverstripe.com/doku.php?id=director#custom_rewrite_rules). Something like:
'filedownload/$Action/$Filename' => 'Filedownload_Controller'
4) Use mod_rewrite to rewrite all file requests for your previously created subfolder (see 1) to your filedownload controller. Something like:

... and you should be up and running.
The beauty of this solution is, that once you decide to disable the logging of downloads, all you need to do is comment-out the mod_rewrite rules (and maybe remove the password protection from the file-folder if you did that). All your file-download links will still be valid!

Hope that helps.

Correction:
The rewrite rule above will only work when you remove the following line from your .htacces:

.. or you could pass the filename as query string and then use $_GET['file'] in your controller to access the filename:

On a second thought: Password protection of the files-folder isn't required, since you're redirecting all file-access anyway.