The Black Hills [1] region of South Dakota is geared toward family-style vacations. There are plenty of attractions geared toward children and many of them, like Dinosaur Park [2] and Storybook Island [3] in Rapid City [4], are free. There are lakes to swim and boat and trails for hiking, biking, and horseback riding. There are miniature golf courses and trout to feed. It really is a wonderland for kids.
There are a few things to consider if you are traveling with the kids. First, if you are planning on staying at a bed-and-breakfast, check to make sure that children are welcome there, as in some instances, they are not.
Second, the environment in the Black Hills region is fairly arid, so make sure that children carry and drink enough water to avoid dehydration and wear sunscreen to avoid sunburn.
And finally, keep an eye on the kids when hiking in the higher elevations of the Northern Hills [5]. Very few of the trails have guardrails and some of the drop-offs can be steep and long. This is also true in the Badlands [6]. The soil is loose and the cliffs are high, so make sure the children are forewarned to stay away from the cliff edges.
Links:
[1] http://www.moon.com/destinations/mt-rushmore-the-black-hills/discover-mount-rushmore-the-black-hills
[2] http://www.moon.com/destinations/mt-rushmore-the-black-hills/mount-rushmore-and-the-central-hills/rapid-city/sights/dinosaur-park
[3] http://www.moon.com/destinations/mt-rushmore-the-black-hills/mount-rushmore-and-the-central-hills/rapid-city/sights/storybook-island
[4] http://www.moon.com/destinations/mt-rushmore-the-black-hills/mount-rushmore-and-the-central-hills/rapid-city
[5] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-northern-hills
[6] http://www.moon.com/destinations/mt-rushmore-the-black-hills/the-badlands