For OTA or in-the-clear cable HD, a good capture card will do. For NTSC capture, a used PCI Hauppage PVR-150 is easy to get ahold of, and excellent. When you say, "digital cable," though, that makes me think a cable box, which means you'll have to tune the cable box from the PC, which can get involved.
But, otherwise, it's just too old, and was low-end when it was new. It should play back HD OK (based on using a Turion 1.8GHz), but don't expect 100% smoothness 100% of the time, and 1GB RAM is a serious limitation for just web browsing, these days.
On the bright side, anything from a used Core 2 Duo on up, >=2.4GHz, with 4GB+ RAM, will have the oomph necessary and then some, if you want to be cheap, and anything but an Intel Atom or AMD C-/E-series will have the power, when it comes to new stuff.
Due to the digital cable part, my thinking would be to check out the logistics of tuning and recording, before going about figuring out a new computer.