This article studies the Cauchy problem for the coupled nonlinear Klein-Gordon equations with damping terms. By introducing a family of potential wells, we derive the invariant sets and the vacuum isolating of solutions. Furthermore, we show the global existence, finite time blow-up, as well as the asymptotic behavior of solutions. In particular, we establish a sharp criterion for global existence and blow-up of solutions when E(0) < d. Finally, a blow-up result of solutions with E(0) = d is also proved.