One- and multi-dimensional stochastic Maxwell equations are considered. With additive noise, it is known that such system can be written in the multi-symplectic structure, and the stochastic energy increases linearly in time. High order discontinuous Galerkin methods are designed to satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyzed. The fully discrete methods are obtained by coupling with symplectic temporal discretizations. Discontinuous Galerkin methods are also designed for stochastic Maxwell equations with multiplicative noise, combined with strong Taylor 2.0 temporal discretization. One- and two-dimensional numerical results are presented to validate the theoretical analysis results.