In my adhoc efforts on this, I've found that different add-ons for different browsers can works for different sites. Downloading is only half the problem, as it may come as swf, for example, and then you have to figure out how to play that. Downloadhelper on FFox seems to work for many. BetterChrome, too.
One way is screen-scraping -- play the video and record screen and audio. Camtasia and Screenflow are helpful here. Jing is free, but limited unless you pay. QT can record audio, or video, but not both at the same time.