What happens when the Runner stack is empty?
Your instinct is correct; the runner continues to play with the cards that he/she has in play. 

It would have been nice if the rules clarified this a little, but since they don't specify that anything special happens you carry on playing.

Despite the runner being able to continue playing, the runner can no longer draw cards and therefore becomes prone to getting flat-lined (having to trash more cards than there are in their hand) and losing that way.

This is fairly rare occurrence, since the corporation draws every turn, whereas the runner doesn't necessarily. Usually the corporation would deplete their deck before the runner, ending the game that way.

Related Links: 


http://boardgamegeek.com/thread/853237/depleted-runner-stack
http://boardgamegeek.com/thread/877799/if-the-runner-runs-out-of-cards-in-his-stack-can-h