Quantum key distribution (QKD) protocols based on high-dimensional quantum states have shown the route to increase the key rate generation while benefiting of enhanced error tolerance, thus overcoming the limitations of two-dimensional QKD protocols. Nonetheless, the reliable transmission through fiber links of high-dimensional quantum states remains an open challenge that must be addressed to boost their application. Here, we demonstrate the reliable transmission over a 2 km long multicore fiber of path-encoded high-dimensional quantum states. Leveraging on a phase-locked loop system, a stable interferometric detection is guaranteed, allowing for low error rates and the generation of 6.3 Mbit/s of secret key rate.