A general form of N-dark soliton solutions of the multi-component Mel'nikov system is presented. Taking the coupled Mel'nikov system comprised of two-component short waves and one-component long wave as an example, its general N-dark-dark soliton solutions in Gram determinant form are constructed through the KP hierarchy reduction method. The dynamics of single dark-dark soliton and two dark-dark solitons are discussed in detail. It can be shown that the collisions of dark-dark solitons are elastic and energies of the solitons in different components completely transmit through. In addition, the dark-dark soliton bound states including both stationary and moving cases are also investigated. An interesting feature for the coupled Mel'nikov system is that the stationary darkdark soliton bound states can exist for all possible combinations of nonlinearity coefficients including all-positive, all-negative and mixed types, while the moving case are possible when they take opposite signs or they are both negative. The dynamics and several interesting structures of the solutions are illustrated through some figures.