We present the first quantum key distribution (QKD) experiment over multicore fiber. With space division multiplexing, we demonstrate that weak QKD signals can coexist with classical data signals launched at full power in a 53 km 7-core fiber, while showing negligible degradation in performance. Based on a characterization of intercore crosstalk, we perform additional simulations highlighting that classical data bandwidths beyond 1Tb/s can be supported with high speed QKD on the same fiber.