I would normally always say a batsman should walk if he believed he has nicked it, but seeing as more and more decisions are being taken away from the umpires now, I don't see how any decisions can be left to the batsmen!
If Broad had walked, there would have been just as many people (in the press, twatter etc) saying he should have waited for the umpire's decision as there would be praising him for shoving his bat under his arm and trudging purposefully back to the pavillion with his head held high.
A very similar thing happened to Adam Gilchrest in Australia 10+ years ago against the West Indies. Only he walked. The press went mental at him for not waiting for the umpire....