Studies of random close packing of spheres have advanced our knowledge about the structure of systems such as liquids, glasses, emulsions, granular media, and amorphous solids. When these systems are confined their structural properties change. To understand these changes we study random close packing in finite-sized confined systems, in both two and three dimensions. Each packing consists of a 50-50 binary mixture with particle size ratio 1.4. The presence of confining walls significantly lowers the overall maximum area fraction (or volume fraction in three dimensions). A simple model is presented which quantifies the reduction in packing due to wall-induced structure. This wall-induced structure decays rapidly away from the wall, with characteristic length scales comparable to the small particle diameter.