If you have other dogs in your home when you introduce your Borzoi puppy, you shouldn't have many problems. The Borzoi is tolerant of other dogs and can get along well with them when raised with them. Also, socializing your Borzoi puppy early will prevent him from becoming aggressive or standoffish around other dogs.
In general, the Borzoi is not a territorially aggressive dog. This is a good thing if you have other dogs in your home or have friends who bring their dogs over to your house. Most Borzoi do not feel the need to protect their territory, so incidents regarding territory between the Borzoi and strange dogs are usually avoided. But there are exceptions to every rule, and no two Borzoi are alike. Some male Borzoi have been found to be aggressive towards other male dogs, so be sure your Borzoi is well-socialized before you have another dog to your home.
However, before you allow your Borzoi to have a doggy friend over to the house, or introduce him to the other puppies in the dog park, you should be very aware of his hunting instincts. Your Borzoi might see another dog - especially a small dog - as something he wants to course or chase down. His instincts will tell him to run after the other dog until he captures it, then he will hold the other dog by the neck or throat until his master commands him to let go. The Borzoi is not a dog bred for fighting, so his behaviors should not be interpreted as violent; nor should these behaviors be interpreted as aggressive or dominating. The Borzoi is simply following his chasing or coursing instincts: he sees the other dog as prey.
This can be quite a traumatic experience, both for the other dog and for the dog's owner. If another dog owner sees this behavior, it could easily be misconstrued as aggressive behavior, and the owner could fear for his dog's life. As for the other dog himself, he will likely be terrified, especially if he is a smaller sized dog. Small and toy dogs are the ones most often chased by Borzoi since they closely resemble small game to the Borzoi. The sheer size of the Borzoi is enough to scare a small dog, but another problem may be the injury of the small dog; if the dog struggles when captured, the Borzoi will probably try to hold on, possibly hurting the other dog. This, too, would cause serious problems with the other dog's owner.
The most important thing to remember about Borzoi and other dogs is early socialization. When raised with another dog, the Borzoi will adapt just fine to that dog. If your goal is to make him comfortable with strange dogs, take him to the dog park and obedience classes from an early age. This will help him get used to other dogs and learn to socialize with them appropriately.