The full time-jitter response of a single-photon detector can make a significant contribution to the quantum bit error rate (QBER) of high repetition rate quantum key distribution (QKD) implementations. Although there have been studies into understanding the contribution for single-mode optical fiber coupled single-photon detectors, the contribution of larger, multimode core diameters to the QBER have not been explored in detail. With the growing importance of free-space QKD, which typically use multimode fibers to reduce coupling loss, it is vitally important to understand how the multimode fiber coupling will impact the total QBER. This work studies the impact of the time-jitter contribution to QBER when coupling a commercial off-the-shelf silicon single-photon avalanche diode with various multimode fibers while simulating operating at 1 GHz with empirical measurements taken at 1 MHz repetition rate. It was found that step-index multimode fibers can significantly increase the QBER, while graded-index fibers can provide an QBER contribution similar to a single-mode fiber. The results highlight that there is a significant benefit in using graded index multimode fibers for a free-space QKD receiver, particularly for high repetition rate applications.