We didn’t take The Steve (my dog) with us to Black Dragon or Little Wild Horse because we were with friends and car space was at a minimum.
There’s a dryfall along the wash to the entrance of LWH that may be an issue if your dog, like mine, isn’t good at scrambling. (It’s easy enough for us bipeds, though). The Steve is a pretty big dog — about 70 lbs — so I don’t really enjoy hauling up and over rocks, but it can be done. If your dog is smaller, or a better climber, I’m sure it can be done.
Here’s some good information on getting to, and hiking through, LWH:
http://climb-utah.com/SRS/lwh.htm
The ranger at Goblin Valley SP said there’s no crypto in the area, but you’ll want to keep an eye out, and make sure if you see any that your dog and kids don’t trod all over it.
LWH is pretty popular these days. I’ve heard reports of the parking area being full on most weekends and some weekdays, so it may not be the best choice if you’re looking to escape the crowds.