The precipitating agent is BaCl2 right? I was guessing the purpose of HCl here. Is it to acidify the pH to allow easy solubility and thus slower precipitation? But even if it is, why would we want a slower precipitation when we can speed by the process? Is it due to that lesser compounds would precipitate at a lower pH, thus reducing error in coprecipitation?

There is a gas in our atmosphere (low, but significant concentration) that will form a precipitate with alkalki and alkaline earth metal ions. What is it? The additon of HCl will expel this gas, or at least ensure that it's conjugate base is not present. The pH will have no effect on the rate of precipitation of BaSO4. It will, as you corrrectly suggest, avoid contamination coprecipitation.