We were left at an impasse on the episode regarding the part of the argument from design referring to the fine-tuning of the universe to support life. Dawkins didn't give enough detail about this for us really to understand, much less critique it, yet it seemed like a lot of what we were concerned about hinged on this argument. You can read about it on Wikipedia.
Prominent in the Wiki article is one of the lesser known among the new atheists, Victor Stenger. The video below shows him talking about this issue. The fine-tuning discussion starts runs from around minutes 16-38. Before getting into the technical details of what fine-tuning amounts to, he first makes the point that if the universe was designed for life, we should expect to see a lot more life in it (less lifeless space and time without intelligent life). We'd also expect more accessible planets to be conducive to life than just Earth (the point being that most of them sure aren't, and even if there are any, we couldn't get to them in a lifetime of travel). Neal Degreasse Tyson makes the same point more energetically here.
The values that allegedly, if different, would prevent life include the speed of light, Planck's constant, the ratio of electrons to protons, ratio of electromagnetic force to gravity, expansion rate of the universe, mass density of the universe, and the cosmological constant). I'll let you watch the video for the details of Stenger's response, but the upshot is that the most important of these constants are self-regulating, meaning they'd approach that rate regardless of where they started. According to Stenger, everything looks exactly as it would if the universe came from nothing.
Personally, I don't feel comfortable enough with the physics either before or after this lecture either to actually see that there's a problem and to see that Stenger has solved this for us. I don't have a sense of the scientific consensus, and Stenger says that some of what he's saying is still a matter of debate among physicists. In conclusion, I sympathize with Dawkins's seeming inability to thoroughly describe this problem or what's wrong with it.