I had this same issue (and asked myself why anyone would use this over rsync), and then found the answer. It is extremely non-intuitive, but you must download the Aspera Connect browser plugin, and the CLI utility ascp is included in that package. If you still have issues, I have it in my Dropbox.
Update: Aspera now maintains this shell script as a separate file. You should now be able to do
The Aspera support responded to the Biostar information request email directly so I am forwarding it here
To Use Aspera. Navigate to http://www.ncbi.nlm.nih.gov/sra/ and click
on 'Download' (left column). You will get a page with a link to Aspera
being available at Aspera Connect under the download tab (if it isn't