Do trainers who use positive-style methods ever use the word "No" to a dog?
Most actually do! Of course there will be times when you want the dog to stop doing something, and there will also be times when you NEED the dog to stop doing something that's dangerous! So let's talk about "No."
First off, the more highly-skilled you are at managing your dog, the less you will find yourself going "No, no, no!" If you find yourself saying "No!" more often than you find yourself praising or rewarding, something is likely wrong with either your training approach or your set-up in general. For example if you are trying to teach "Stay," and the dog keeps breaking -- you probably are rushing too fast and lumping in too much distance, too much time, etc. If you are trying to housebreak a puppy and find that you are constantly getting after the puppy for accidents, then you should probably fix your schedule or supervision habits with the puppy. Good trainers are able to set the dog up to succeed most of the time, good habits develop easily, and the dog never even thinks to do the wrong thing. This is ideal!
But even good trainers might want to communicate to a dog that it's doing the incorrect thing (lying down instead of sitting on the command "sit," for example) or need a dog to immediately stop doing something (jumping up on an elderly relative for example.)
There are three basic types of "No."
The first "No" is "No, you're not doing it right." Some trainers call this the "No-Reward Marker." If you say sit, and the dog lies down, and you say "No," then the dog understands to try something else -- hopefully a sit! The No-Reward Marker is sometimes "No," but frequently people will use "Nope," or "Oops," as this has less chance of making the dog feel bad and simply give up and stop trying. I use "Oops" for passive behaviors (Stay or leave it) and just silence for harder or more complex behaviors.
The second is "No, it's not going to happen, so might as well give up." If you are eating a bagel or something, and your dog wanders up to you "Can I have some bagel?" then the answer is "No." The most important thing here is to stick with it! If you are consistent enough, then the dog will give up and stop whenever it hears that word! I usually just shake my head "no" in this kind of situation with my dogs, and they get it!
The third "No" really means "Stop." As in "STOP WHAT YOU ARE DOING RIGHT NOW." This is the kind of No that people mess up most frequently. First, they overuse it. "NO!" because the dog is doing one thing. "NO!" because it's doing another thing. "NO!" because of something else. Continued and repeated use of "NO!" has a tendency to desensitize the dog to the sound. You will find that you are escalating and shouting louder and louder at it and even that's failing to work. The second most common error is that if you use this "No" then you MUST ensure that the dog stops what it's doing. It is unhelpful to sit on the couch shouting "NO!" as your dog hops up onto the kitchen counters and eats all of the turkey. You MUST be prepared to get up and enforce. The third most common error is failure to give the dog a consequence. If you use this "No!" and the dog responds (stops doing what it was doing) you must immediately tell the dog what it should be doing instead. For example, if you "No!" to get the dog to stop jumping on your relative, now you should tell it "Sit." (Use just praise and not food treats for this sit, BTW -- you don't want the dog to learn the pattern of Jump, "No!" Get off, "Sit," Sit, Cookies.) If you use "No!" and the dog does not obey, you must give it some sort of consequence. Time-outs or long downs are usually effective. I usually use the word "Stop" for this version of "No," as I find that it makes the voice stronger and more effective, especially for female handlers.