I don't think the general presence of dogs has a big impact on deer in general outside the season. That is not to say that if fido's favorite hangout was Mr. Big's bedding spot, and it is frequent enough, that Mr. Big won't find a new place to bed which may or may not be on your lot. In the season, dogs just add one more thing to the pressure factor that deer feel.
Deer dogs are a different story. Unfortunately, my place is in deer dog country. While dog hunting for deer is not my cup of tea, I've got no issues with dog hunters as long as they hunt on parcels large enough that the likelihood of their dog entering unauthorized property is low. I do understand that dogs can't read posted signs, but if there is a significant chance they will encroach on the lands of another, they parcel they are hunting is not large enough.
My big issue was with dog hunters intentionally releasing their dogs along our land and going around to the other side to shoot at deer that have been chased out (or sometimes sneaking in). Our state has laws that are very favorable to dog hunters allowing them to retrieve their dogs on posted land without landowner permission.
We also have domestic dogs that come run on our land from time to time. Some don't chase deer, but others pursue them relentlessly. Depending on the stress period, that can be one more stress. This is not the same as taking a bird dog out to hunt quail, a rabbit dog to hunt rabbits, or just taking a pet for a walk.
My approach would be to simply ask the neighbor nicely to keep fido on his property. Let him know that you understand, dogs sometimes get away, but that is different from folks letting their dogs run free. It is not just deer, but other wildlife that can suffer as well depending on the dog.
We did have a neighbor who did not respond well to the nice approach. The encroachment was occurring on a weekly basis. After taking the dog back to the owner a couple of times without the behavior corrected, I began taking the dog to the animal shelter. It had a collar and they would give the dog a health check, feed it, call the owner, tell him to come pick up his dog, and charge him for the care to get the dog back. It was a small fee the county allowed them to charge, but the inconvenience of driving to the shelter and pay the fee was enough to get the guys attention.
Thanks,
Jack