I would think this would depend on where it is being downloaded from - if you mean from the publisher's site then I think you're reliant on their weblogs and what data you can get from them. There is a standard for reporting statistics on download and searches of electronic resources that most publishers (for the UK anyway) conform to, which is called COUNTER. If it's a resource that your institution subscribes to them whoever manages that subscription (probably within the Library) would be able to explain what statistics they have that might help you.
You might also look at the tool Impact Story http://impactstory.org/ Number of downloads is one of the metrics collected.
I am not sure that it can capture all or even most downloads from within subscription databases though it looks like it does capture downloads that use DOI Identifiers and PMID (Pub Med) identifiers.
I can't think of any but, as a new member of ResearchGate, it's a facility that they offer. Of course, it's only a record of when you list it - rather than any past downloading