What types of cables support closed captioning?
Your question was initially confusing because you had focused on "cables" which is not the correct way of looking at this.

Once you've decoded a video signal to components, the closed captions are no longer available. If they were not inserted as open captions (aka subtitles) then they're simply not there.

So any transport that removes the Line 21 caption data or the digital caption packets eliminates your ability to defer captioning at the distant end -- you must insert the captions / subtitles before transport. That's the case with DVI / VGA / HDMI.

If you must allow the ability to choose a captioned or non-captioned display at the distant end, you would have to send two component feeds (e.g. two CAT5s), one captioned and one plain, and A/B switch them at that end.