The highly-regarded US current affairs show (not the satirical Australian current affairs show) Frontline has an archive allowing you to view selected episodes online. Let's hope this inspires the BBC into getting their archives online faster, and with free access to everyone, not just those in the UK.
The NewsHour with Jim Lehrer also has its segments viewable online, with your choice of video or audio only, of the latest program. You don't lose much when choosing just the audio, as unlike some current affairs shows, the story is in the ideas, not the pictures.
The NewsHour site is a bit confusing when it comes to finding their audio and video. You can see what was on each day in the index, which has links to transcripts and RealAudio of some segments. If you take a different route, to their video page, you'll see a shorter list of recent segments available as RealVideo. Even when the same story has audio and video available, the links to these streams are in different places.