She should have called her dog and apologized.
I will say, I was just in a similar situation on Weds, except my dog was on a leash. I have 5 month old black lab puppy, very sweet, and DD 4.5 and I took her for a walk at a park near my house. I hadn't been there with her, but had thought it would be great for her, since she's still a puppy, it's fairly flat and not too large, and has a nice mix of dirt trail into woods that's not too challenging for my daughter.
We were walking on the path and a woman jogged around my daughter, and I when I apologized for DD being in her way (because she was walking like a 4 year old and tough to get around) the puppy jumped up and put her paws on the lady's mid-section. I had the leash slack and just wasn't expecting it. My dog is a puppy, and I don't see how she could be perceived as threatening, but I do see how it could be annoying to have a dog jump up on you as you are trying to run.
Anyway, she let me know that DOGS ARE NOT EVEN ALLOWED IN THIS PARK! And I was like, really? I'm sorry, I didn't know we'll leave. And she kind of spat at me IT'S POSTED ON THE SIGN WHERE YOU WALK IN. And I felt like total crap. I have pms, and I must admit--I actually cried a bit once we got back in the car. She was absolutely in the right but I was still very taken aback and felt embarassed.
I had imagined that at the trailhead there was a giant picture of a dog with a slash through it with No DOGS ALLOWED in red or something similar. Actually, there's a sign that lists the park rules and along with no alcohol, nobody in the park after dark, and no firearms, dogs are also listed. And the the kicker--right next to the sign, is one of those disposable doggy poop bag dispensers.
You're absolutely not in the wrong, and as the mother of a four year old and as a dog owner I don't think you're unreasonable to not want unleashed dogs invading your space. I guess all I'd offer is what you've already posted in follow up--that in the future maybe treating the offender with some gentleness would make for a better day for everyone, you wouldn't feel like someone thought you were a "dog-hater" and they might be able to realize that they are actually in the wrong.
You asked about a different way to handle, and I'd just go for honesty. "You know, we actually love dogs, I used to have 5. But I can tell you, a lot of children would be very afraid of a strange, large dog charging up to them. Also, dogs aren't allowed on the beach, so parents shouldn't have to watch out for them here."