Keep in mind... Mt. Rushmore "area" (Sturgis, Custer, Deadwood & Rapid City) will be crazy with motorcycles from about 7/20 thru 8/10, due to the 75th Anniversary of the Sturgis Rally..
If you're ok with it (motorcyclist are generally great people).. it would be a neat sight to see and you'll be assured there will plenty of "action" going on..
We actually did Mt. Rushmore and Yellowstone (two separate years) the week before the Sturgis Rally and it was fine.. I stayed in Custer and the site fee marginally went up for the rally.. this year may be a bit different, seeing as it's the 75th... but Custer is over 70 miles away and not in the heart of the Sturgis debauchery
There are plenty of "bikers" who are out and about even as far away as Yellowstone the week prior to the Rally.. again, these are good people, having a good time..
Have fun.. those two area's are awesome and you're lucky to have the time necessary to see them...
Good Luck !!