As one of the most pressing issues in the world, climate change has brought significant impacts on natural and human systems, where drought events are likely to be frequent and severe under climate change impact. In this study, a multi-GCMs Bayesian copula (MGBC) method is developed for assessing the impact of climate change on drought risk, through integrating multi-GCMs, Bayesian, and copula function into a general framework. MGBC can tackle uncertainties existing in copula parameters and global climate models (GCMs), as well as provide ensemble projections for drought risk. MGBC is applied to the Aral Sea basin for assessing multivariate drought risk (i.e., duration-severity, duration-peak, and severity-peak) during 1901–2017 and 2021–2050, where climate data are derived from multi-GCMs (CMCC-CM, GFDL-CM3, MRI-CGCM3) under RCP4.5 and RCP8.5. Some major findings can be summarized: (1) in the historical period (1901–2017), the basin suffered 119 drought events, where drought risks of middle and lower reaches are higher than that in the upstream; (2) in the future (2021–2050), the basin’s drought risk level will increase (1.5%∼8.6%), particularly for middle and lower reaches due to climate change impact; (3) as the impacts of climate change intensify, drought risk would increase (i.e., RCP8.5 > RCP4.5), and drought events in association with high-risk level would move from the basin’s south to north; (4) the main factors that bring the drought risk are meteorological condition (e.g., precipitation and evapotranspiration) and human activity (e.g., soil moisture and texture). The findings suggest that droughts in the Aral Sea basin are affected significantly by climatic factors and drought risk would intensify with climate change.