Camping is allowed in most forest areas. But in a popular forest like the White River, it's especially important to be kind to the land. Don't build any new fire rings; better yet, use a camp stove. Camp well away from lakes and streams. And you know to pack out your trash.
If you prefer a developed campground, the Dow Campground guide is an excellent reference. Here are some good ones culled from their pages...
Camp Hale Memorial - 21 campsites on the grounds of historic Camp Hale. Holy Cross RD.
Weller - 10 campsites located conveniently near Aspen. Aspen RD.
Cutthroat - 14 campsites with excellent views of the Flattops Wilderness. Blanco RD.
Yeoman Park - 22 campsites. Excellent wildlife-viewing opportunities nearby. Eagle RD.
Deep Lake - 36 campsites located on a flat top mountain "far from everything." Super views. Eagle RD.
Gold Park - 10 campsites in a "sweet little campground." Holy Cross RD.
Bogan Flats - 35 campsites stretched along the Crystal River in Aspen Valley. Sopris RD.
Gore Creek - 22 campsites near Vail. Holy Cross RD.