If the browser recognises the file type then the browser will process the file itself rather than offering the file for download. Files offered for download can be saved to where ever you choose or opened and processed by any program on your computer. Those not offered for download are downloaded to the browser cache and processed by the browser itself making it harder for the less computer literate to actually save a permanent copy or run it in a different program.
You can just put a link to it. If they click it and are set up as Stephen describes then they'll play it. If not then they'll download it and play it from there. In either case they can right-click the link and choose to save it rather than play it.
"Debugging is twice as hard as writing the code in the first place. Therefore, if you write the code as cleverly as possible, you are, by definition, not smart enough to debug it." Brian W. Kernighan
you can also just .zip the file that you want people to download and make a standard link to it. Zip files will always ask to be downloaded to the user's computer.
So for your client, you may want to have a 'listen now' function that links to the actual file, or a 'download it' function that links to the zip.