Would the Mac mini or Macbook Air have better sound hardware for speech recognition?
I believe I am correct in saying (without a Mini to cross check against) that they both (along with all the Mac lineup) use the integrated Intel HD audio chip, and as such there should be no real difference between the 2.

I would say that you limiting factor would be that the Mini does not have an integrated microphone, whereas the Air does, so you will need to bring your own mic.  Also, neither have separate mic input/audio output ports, and require you to use something like an iPhone headset (modified 3.5mm audio connector with both channels on the same connector), which may not provide the quality (or comfort) you need.

So, on the Mini, you will need to buy a headset in order to get mic input, and if you already have one you may need to replace it with a USB one or iPhone one.  On the Air you're good to go unless you are already a regular headset user, in which case you may beed to replace if it uses 2 discrete connectors for headphone/mic.