Dart is excellent in few cases on the client. First, if you can forget about older browsers, and second if you don't need to use a lot of pre-existing JavaScript libs in your app, and third if you don't need to use the web control on iOS. Dart as a Chrome Packaged App should shine. I really enjoy developing in it. I don't have any experience with it on the server, so couldn't say there.
That is actually the only way I see any future for Dart.