First, let's review some basic terminology.
Tuner: A tuner is a device that allows you to find and change channels. Cable and satellite boxes typically have tuners. Television sets also have tuners.
TV Display: A TV monitor or display provides the television view. Sometimes, particularly in the digital world, they provide only the display function and do not have tuners. For example, you can hook up a digital cable/satellite box to your old analog TV. The TV then just becomes the display, with the box providing the tuner service.
Closed Captioning Control: Devices with tuners typically control the closed captioning function. So when you have a cable or satellite box, usually you use the box's menu to turn closed captioning on or off. (If it is turned on, and does not show, then you may have to also enable it via the TV's menu.)
Closed Captioning Data: When networks send the TV's picture to your house, the closed captioning arrives as a separate piece of data that must be decoded by the tuner. In the old, pre-digital television days, that was always done by your television set.
Now let's talk about the HDMI cable. For some reason, when the engineers created the cable's specifications (to ensure that all cables labeled "HDMI," regardless of manufacturer, would behave the same), they left out any requirement to carry closed captioning data.
As long as your TV's setup ensures that your cable/satellite box connects directly to your television, the box's tuner should decode the data and send a video signal that includes the closed captions (handled by the HDMI cable as open captions as part of the video signal) to your monitor.
(If you receive your HDTV signal over the air, your HDTV should decode the caption data just fine.)
However, if you run your cabling system through a DVD player or DVD recorder, you might lose your captions. The same goes for DVDs with closed captions - if the DVD is connected to the monitor via HDMI, the captions will not appear.
Why is that? Because most DVD players/recorders do not have tuners and therefore typically do not pass on closed captioning data. Some do, but not many.
If you want to integrate a cable/satellite box and DVD player or recorder in your setup, your best bet is to employ an a/v receiver to manage your connections so that the cable/satellite box and DVD player each have a direct connection to your monitor.
Until the HDMI specs change, if you want to use HDMI cabling between the DVD player/recorder and the monitor, you will have to rely on subtitles rather than closed captions.
For more information, the CEA (Consumer Electronics Association has written a nice, in-depth article about why captioning works differently in hi-def and other digital television setups.
To register a complaint about current HDMI specs, contact the HDMI licensing consortium: http://www.hdmi.org/contact/index.aspx (link opens in new window).