You wouldn't address it to the base itself.. any correspondence you sent would be addressed to the person, and the unit they are in would be included. The mailing address would look something like:
Joe Private
C. Co., 1/999th IN
Fort Jackson, SC 29207
The second line would be their unit, and you would need to obtain this information before you posted anything to them. Remember that an Army post will typically have 10,000 or more people, and it's too much work to simply try finding personnel by name alone, especially on a post like Fort Jackson, where soldiers are constantly coming and going as it is one of the posts where Basic Training is conducted.
The web address of the Fort Tabor Military Museum is: http://www.forttaber.org
The address of the Fort Lewis Military Museum is: Main St, Fort Lewis, WA 98433
Fort Jackson South Corolina
The address of the Fort Stevens State Park Military Museum is: 1900 Ocean Drive, Hammond, OR 97121
The address of the Fort Douglas Military Museum is: 32 Potter St, Salt Lake Cty, UT 84113-5046
The address of the Fort Snelling Military Museum Volunteer is: 511 Constitution Ave, Fort Snelling, MN 55111-4018
The address of the Friends Of The Fort Lewis Military Museum is: Po Box 331001, Fort Lewis, WA 98433-0009
The web address of the Augusta Military Academy Museum is: http://www.amaalumni.org/museum/
The address of the Augusta Military Academy Museum is: 1640 Lee Hwy, Fort Defiance, VA 24467
The address of the Organization For Texas Military Education is: 712 Dorothy Lane, Fort Worth, TX 76107-2039
Fort Jackson, SC is about 20 miles closer to Statesville than Fort Bragg/Pope AFB, NC.
The address of the Fort Tabor Military Museum is: 1000 S Rodney French Blvd, New Bedford, MA 02744