Underneath the list of all caption sets for a project, you'll see a link to 'Download all'. Clicking this will offer a list of file formats, and having chosen one, you'll download a zipfile of all valid caption sets for a project, in the format of your choice.
If a caption set still has errors, it won't be included in the zipfile.
If a caption set is valid, but unapproved, its filename will indicate that.
You can also download individual caption sets from the 'download' link that will appear alongside the caption set, providing it is valid.