Revealing the impact of future climate change on the characteristics and evolutionary patterns of meteorological and hydrological droughts and exploring the joint distribution characteristics of their drought characteristics are essential for drought early warning in the basin. In this study, we considered the Jinghe River Basin in the Loess Plateau as the research object. The standardized precipitation index (SPI) and standardized runoff index (SRI) series were used to represent meteorological drought and hydrological drought with monthly runoff generated by the SWAT model. In addition, the evolution laws of the JRB in the future based on Copula functions are discussed. The results showed that: (1) the meteorological drought and hydrological drought of the JRB displayed complex periodic change trends of drought and flood succession, and the evolution laws of meteorological drought and hydrological drought under different spatiotemporal scales and different scenario differ significantly. (2) In terms of the spatial range, the JRB meteorological and hydrological drought duration and severity gradually increased along with the increase in the time scale. (3) Based on the joint distribution model of the Copula function, the future meteorological drought situation in the JRB will be alleviated when compared with the historical period on the seasonal scale, but the hydrological drought situation is more serious. The findings can help policy-makers explore the correlation between meteorological drought and hydrological drought in the background of future climate change, as well as the early warning of hydrological drought.