Download and install the latest version of MediaHuman YouTube Downloader. It's available for Windows, Mac OS X and Ubuntu/Linux.
Navigate to the playlist you want to download in your favourite browser and copy its link to the clipboard. It can be a channel, a playlist or a video contained in a playlist. In our example it's a video from one of the VEVO's playlists.
Run YouTube Downloader and paste the link by pressing the first button on the toolbar or using paste hotkey (Cmd-V on Mac, Ctrl-V on Windows and Linux). You can also drag'n'drop it from the browser.
If you added the link to the video contained in playlist, click on Add Playlist icon. And all the videos from a playlist will be parsed, added and ready to download.
Simply click the Start all button to begin downloading all the videos from the playlist you've just added.