If you were just doing SD then you could do the box in the basement and use a MediaMVP to stream the video. However, there aren't any really good options for this for HD yet. There are HD options, but none of the integrate into the pvr software for good WAF.
If you're building a computer to go in the basement for HD anyway, it won't take that much more to put it next to the tv and this is probably your best bet. Here's a couple of things to consider:
A nice looking case. Silverstone makes great cases for this that are affordable too. I like the LC20.
Quiet fans. This means case fans, power supply, and for your cpu. I don't use fans on my videocard and go with a passively cooler videocard.
A good remote. This might cost you the most depending on how fancy you go. The Logitech remotes are nice and they go from $50-$300.
Check out this build I did for my parents. It ended up pretty quiet and if you don't go crazy on all the parts like my dad did, you could keep the cost down too.
http://forum.byopvr.com/dvr/index.php/topic,7453.0.html