A key step in solving minimax distributionally robust optimization (DRO) problems is to reformulate the inner maximization w.r.t. probability measure as a semiinfinite programming problem through Lagrange dual. Slater type conditions have been widely used for zero dual gap when the ambiguity set is defined through moments. In this paper, we investigate effective ways for verifying the Slater type conditions and introduces other conditions which are based on lower semicontinuity of the optimal value function of the inner maximization problem. Moreover, we apply a well known random discretization scheme to approximate the semiinfinite constraints of the dual problem and demonstrate equivalence of the approach to random discretization of the ambiguity set. Two cutting plane schemes are consequently proposed: one for the discretized dualized DRO and the other for the minimax DRO with discretized ambiguity set. Convergence analysis has been presented for the approximation schemes in terms of the optimal value, optimal solutions and stationary points. Comparative numerical results are reported for the resulting algorithms.