You can now use the YouTube API to access and create captions for your videos. That means you could let your site visitors refine the transcript to videos you own, or create their own versions of the caption.
YouTube's short announcement points to its caption developer guide. You need to be able to have authorization to access videos, which most likely means the videos you can access would be limited to your own. Still, with the proper structure in place, you could allow others to change your own captions.
For an example of closed captioning, see this video of Kevin Marks talking about OpenSocial and social applications (find the closed caption control, "CC", via the up arrow control in the lower right)
More likely might be using another service to auto-caption the video. For example, you might link Mechanical Turk tasks to your video paying pennies to actual humans to transcribe your video (Mechanical Turk API profile). Or, perhaps you'd export the audio and use SpinVox (SpinVox API profile), whose popular paid API we covered recently.
To be clear, YouTube hasn't created a crowd-source-ready framework but these useful additions to the API can give developers the pieces to put one together. Those with massive amounts of video would be well-served to put the work in to make this process easy.