I think it depends a lot on which breed you're talking about. Taken as a whole, I would argue that dog shows have been good for dogs. For breeding purposes, it is extremely helpful to have a knowledgeable, impartial judge say "Yes, this dog is an excellent specimen of the breed". It makes it a hell of a lot easier for dog buyers to sort out which breeders are worth their time and which aren't (assuming, of course, they are looking for reputable breeders in the first place).

Additionally, in breeds with distinct working and show lines, there tends to be a noticeable temperament difference between the two types. Working dogs are usually higher energy, higher drive, and more likely to need a job. For people that want "just a pet", show line dogs may be a much better option than working dogs.

BUT, in our efforts to create dogs that breed true, we have also cemented a wide variety of horrifying health problems into a number of breeds. Some breeds are more affected than others, but I think it's irresponsible to deny that inbreeding has increased the prevalence of genetic diseases in dogs. Fortunately, our knowledge of genetics has increased exponentially over the last 20-30 years, and there are now a number of tests that can be done to determine if a dog is likely to pass health problems on to their offspring. Today, no responsible breeder would mate dogs that do not have the appropriate health clearances for their breed. Of course, there is still a lot we need to learn and much work to be done, but I am optimistic that we will see a decrease in the number of genetic diseases in the future for most breeds.

Now, there are some breeds that I do think have been hurt by the dog show world. For most of these breeds, it's been an issue of selecting for more extreme features. More extremely sloped or roached backs in German Shepherds and more extremely pushed-in faces for the brachycephalic breeds come to mind. The breed clubs and breeders of these dogs really need to make an honest assessment of the state of their dogs. Pedigree Dogs Exposed is much maligned on /r/dogs, and to a certain extent, I do think that is deserved. I have seen both that documentary and its follow-up, and I think they are often one-sided and occasionally sensationalist. However, there are some valid concerns presented. In my opinion, certain breed clubs' reluctance to admit the problems with their breeds is one of those valid concerns.

As for the whole "mutts are healthier" thing, I think people vastly overestimate the power of the mutt. Sure, "true" mutts may be slightly healthier than purebred dogs when you look at both populations as a whole, but it's not like every purebred dog dies at a young age of horrible diseases, while every mutt lives a long, healthy life. Also, that lab/GSD cross? Not a "true" mutt, and probably just as prone to hip dysplasia as both of its parent breeds. I'm looking at you, all you doodle owners that claim your dogs are healthier because they're crossbreeds.

tl;dr: I think dogs shows have helped dogs as a whole, but have hurt a few individual breeds. Hopefully, advances in genetics will help all dogs live longer, healthier lives in the future.