As a long time user of [ link redacted, please do not post links in the reviews section ] for presenting youtube playlists for clients as well as a [ link redacted, please do not post links in the reviews section ] I host, I got really tired of the frequent updates which broke the plugin so often that manual roll-backs to previous builds were necessary, and one dreaded to update the plugin at all.
The other popular plugin, [ link redacted, please do not post links in the reviews section ], is absolutely broken when it comes to playlist handling (though its player is responsive).
Enter - a generically named "WordPress Video Plugin".
Holy cow, this thing spanks the pants off both those other plugins. Not only is it robust (works flawlessly for my WP 3.8.1 installations), reliable, flexible (in terms of embed services), but is also responsive right out of the box.
This is a gem which I will whole-heartedly recommend as well as donate to.