Peak One has all the amenities you need - water, (really) nice bathrooms with running water, and a fire pit with a picnic table. A simple campground that would be great for kids, Peak One is safe, comfortable, and a great family spot.
A great location right by Vail, Gore Creek Campground has a great setup - private, clearly defined campsites, fire pits, picnic tables, and bathrooms. No running water though so stop by the Safeway in West Vail and pick up a couple gallons. Great place to camp, felt safe and comfortable. Only downside for me is the consistent hum of the highway not far away.