Banshee doesn't currently have the option to save album art to a track's metadata. For more information, you can subscribe to Bug 600885.
Instead, Banshee stores art in
~/.cache/media-art according to the Media Art Storage Spec. Theoretically, if other media players followed the same specification, they would also have access to the same album art, but as far as I know, Banshee is the only media player that has implemented support for this spec.
Instead, you may wish to use one of Banshee's Community Extensions, banshee-extension-albumartwriter, which allows you to save the album art as a
cover.jpg file inside the folder containing the audio file. This isn't quite what you're asking for, but it's much more widely supported by other music players, and it has the benefit of not duplicating the image file inside each individual track.
If this solution doesn't meet your needs, your only option may be to use another tool to do the job.