Maybe I'm biased, but I vote for Professor Stokes as the most adept at handling supernatural emergencies.
Mrs. Johnson would be a very close second. Her common sense kept her grounded while the rest of the family was often caught up in some sort of paranormal mess. If ever she encountered something out of the ordinary, (e.g. Quentin's ghost) her reaction was always believable and never over the top. Harry wasn't on the show for very long and wasn't really exposed to the amount of weirdness that the other characters faced. Julia did see a lot during her extended stay at the big house, and at times rose admirably to confront whatever bizarre force had entered the Collins' midst (e.g. Nicholas); other times, she would go to pieces emotionally, fretting and crying to Barnabas.
Stokes knew the most about the occult, and his background proved invaluable on many occasions. He was intelligent and brave, able to keep his cool and refusing to be intimidated by the unknown. When the other characters were confused or frightened, Stokes always seemed to know what to do. Of the four, he is the only one who doesn't live at Collinwood, and I will admit that that probably gives him an advantage. Nevertheless, I still think he's the most capable candidate.
ProfStokes