I've lived in Asheville 20 years, and I have been inside the house once. I only did that because my father was visiting and wanted to see it (and was paying). Quite honestly, I wouldn't go back inside even if it was free.
From time to time, friends have given me free passes, which I have used to tour the gardens and grounds. Those are beautiful, but still not worth spending more than a few bucks on.

Asheville is surrounded by so much natural beauty which you can enjoy for free. Visit the Smokies (1 hour away) or cruise the Parkway. Or take the kids to the Western NC Nature Center, to Sliding Rock, go whitewater rafting, to the Colburn Earth Science Museum, the Cradle of Forestry (which is near to Sliding Rock), ride on the Great Smoky Mountains Railroad, etc..........

Tell us more about your family's interests and more suggestions can come.