The coupled Korteweg-de Vries (CKdV) equation with self-consistent sources (CKdVESCS) and its Lax representation are derived. We present a generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the CKdVESCS as well as the formula for the N-times repeated GBDT. This GBDT provides non-auto-Bäcklund transformation between two CKdVESCSs with different degrees of sources and enables us to construct more general solutions with N arbitrary t-dependent functions. We obtain positon, negaton, complexiton, and negaton-positon solutions of the CKdVESCS.