Gale crater's central sedimentary mound (Aeolis Mons or, informally, Mount Sharp) preserves one of the best records of early Martian climatic, hydrologic, and sedimentary history. Mount Sharp's sediment sequence is broadly consistent with a transition from wetter surface conditions with lakes and fluvial activity, to groundwater influenced sediment cementation under dryer conditions, to a period of anhydrous sediment accumulation and erosion dominated by aeolian processes. However, surface conditions alone do not provide a direct constraint on the climate evolution. We use models of the hydrological evolution of the crater, evaluated against constraints on the local depositional environment from both satellite observations and ground-based observations by the Mars Science Laboratory on the Curiosity rover to shed light on the evolution of the climate at Gale crater. The results show that the local depositional environment is influenced by both climate and the state of crater infill. The overall trend in the stratigraphic sequence, recording a transition from wet to dry environments within Gale crater, could have formed under a dominantly arid climate with changes in sediment properties exposed along the edge of Mount Sharp resulting from both lateral and vertical changes in depositional environment due to a decrease in the accommodation space and changing basin geometry as the crater filled with sediment. Alternatively, the sedimentary sequence within Gale crater could be explained by a changing climate, with lake formation during the waning period of a wetter epoch followed by secular drying to account for the observed phyllosilicate to sulfate transition. In either scenario, short-term climate changes are required to explain local changes in facies and mineralogy. A later shift to much dryer conditions is required to explain the aeolian erosion of the mound, followed by renewed arid to semiarid conditions to explain the later-stage lakes. Hydrological modeling constrains the early climate at Gale crater to have been between arid and semiarid, largely consistent with geomorphological and geochemical constraints on the climate at the Noachian-Hesperian boundary.