The fundamental argument they are having is whether or not deep thinkers learn to be deep thinkers or if they are born to be deep thinkers. If thinking deeply is a learned behavior, then Carr may have a good argument. Then you move on to the specifics of whether or not the Internet promotes skimming or thinking deeply (my opinion is it depends greatly on where you go on the internet). If deep thinkers are born that way, then it doesn't matter.
The argument seems more subtle than that. Carr thinks that deep thinking is learned (or at least, promoted) through old methods of media consumption, but that our new methods of consumption are ruining this ability. Pinker also thinks that deep thinking is a learned behaviour, but that it is taught (and learned) in the institutions where it is most needed, in particular in universities.
Pinker's not worried about recent changes, because he's confident that people who need these skills pick them up, and uses increasing success in sciences as evidence that nothing is going too wrong. Carr doesn't believe this evidence is sufficient, since he believes that modern science may not need deep thinking for its advances. That claim seems to severely underestimate the difficulty of doing good science, or even average science, and seems trivially false.
Really though, Carr values "deep thinking" in and of itself, and doesn't care if people who need it can do it. He's worried that the general population as a whole will not be able to think deeply on anything, but instead will become light "skimmers" of information. It seems to me that the ability to skim and critically combine information from multiple sources is incredibly important now, maybe more important than the "deep thinking" Carr promotes.
I definitely side with Pinker here. The skills are always around for those who want or need them. Nothing about our current consumption habits prevents us from learning them or using our self-control and employing them. Carr should be deeply uncomfortable with the amount of information we need to wade through day in day out, and realise that people are just adapting to do the best they can in our modern environment.