Yes, you will still get HD via component video cables (component can carry up to 1080i, which is the plenty for HD broadcasts). I've used both DVI and component with my HD digital cable box, and I wouldn't worry a bit about switching from DVI to component for that.

If you did decide to stick with DVI for both sources, you would probably be looking at something like this or this from Gefen. You may find a somewhat lower price elsewhere online or a different manufacturer (although Gefen has a huge share of that market), but you're still looking at a couple hundred dollars.