From Dropbox Community dated 04-08-2015 08:17 AM:
Q: how do I track downloads of any given file that I've uploaded to my account?
A: You can't, Dropbox doesn't provide that feature.
There may be workarounds however. An insecure site has offered two:
1) Short URL services like Bitly and Goo.gl will track how many times
the links are clicked, showing you the count of how many times your
file has been clicked on.
2) Although Dropbox doesn’t have a downloaded link count, it does have
the Dropbox API which allows developers to build plugins to help you
do that.