I am having the same problem with my male puppy. He has jsut recently experienced rain and I started picking him up and taking him to the grass but I am afraid I reinforced him for not going out. Any thoughts on carrying vs. literally having to drag him out to an appropriate spot?
Well, I'm not sure on this one, as I've never had a Basenji puppy… I adopted mine as a crotchety 7 year old (now a crotchety 12 year old) However, if my boy is any indication of the breed, I'd say that picking up your pup is not a problem. Mine seems to loathe being picked up, he'll tolerate it, but it's certainly not a reward for him to reinforce any sort of behavior. In fact, I think mine would start going out on his own just to avoid the humiliation of being carried. For my B, being picked up has the connotation of the "argument" being non-negotiable - if he will not go where I say on his own, I will just put him there (calmly and gently, of course).
Additionally, as others here have stated, when your dog does his business in the rain, he gets rewarded for doing it quickly by getting to come back inside where it is dry. My problem has been that the weather reinforces my dog to "hold it" since the desert rain inevitably lets up before he goes.
One more tip - if you have any gravel, I have found that a lot of dogs that dislike water prefer to potty on gravel over grass when wet.