I have some jQuery tabs one of which holds a flash video. When I play the video in one tab and click to another in FF or Safari the video stops along with the sound, clicki
The most helpful thing you could do is the post which video player you're using. Youtube and Brightcove both have Javascript APIs that you can use to control the player. Like tahdhaze says here, the best thing to do would be to stop the player on the window.onBlur event, and start it again when it gains focus.
Using CSS to stop the video from playing is at best tricky, at worst a hack... use the Flash-Javascript bridge API that comes with the player.