A system of magnetic dipoles in two-dimensional (2D) channels was studied using Brownian dynamics simulations. The dipoles interact with a purely repulsive r(-3) potential and are confined by two hard walls in one of the dimensions. Solid crystals were annealed in the 2D channels and the structural properties of the crystals were investigated. The long-ranged nature of the purely repulsive dipoles combined with the presence of hard walls led to structural deviations from the unbounded (infinite) 2D dipolar crystal. The structures in the channels were characterized by a high density of particles along the walls. The particles along the wall became increasingly localized as the channel width was increased. The spacing of the walls was important in determining the properties of the structures formed in the channel. Small changes in the width of the channel induced significant structural changes in the crystal. These structural changes were manifested in the density profiles, defect concentrations, and local bond-orientation order of the system. Oscillations in the structural properties were observed as the channel width was increased, indicating the existence of magic-number channel widths for this system. As the channel width was increased the properties of the confined system approached those of the unbounded system surprisingly slowly.