As of version 2.9.17, Downcast for Mac can finally be controlled using an Apple keyboard's media keys. Since Apple still doesn't permit sandboxed apps to control the media keys, we have to do this with the help of a free 3rd party app called BeardedSpice. This app acts as a system wide controller for your media keys, passing media key press events to the appropriate application whether it's Downcast, Spotify, iTunes or some other app.
For more info on BeardedSpice and for install instructions, head over to their website and download the latest version, which includes support for Downcast.