Laya, just a thought, but you may want to try desensitizing Justin to thunder. Why does he dislike it, do you know? Was he maybe left out in it for an extended time? (I had a Belgian Trevurn that was totally petrified of it, but he had been wandering in a severe storm when I found him). The thing is, most horses take it in stride (please forgive the pun, he-he!) and you don't want to be at a show and have him freak out on you; (most shows around here continue on despite the elements, unless there is also severe lightning...you know, "the show must go on!!")
As far as your original question, drizzle is fine to ride in, as long as the footing isn't too sloppy, like the other posters stated.
HL2