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Traditional univariate drought indices may not be sufficient to reflect comprehensive information on drought. Therefore, this paper proposes a new composite drought index that can comprehensively characterize meteorological and hydrological drought. In this study, the new drought index was established by combining the standardized precipitation index (SPI) and the standardized baseflow index (SBI) for the Jiaojiang River Basin (JRB) using the copula function. The prediction model was established by training random forests on past data, and the driving force behind the combined drought index was explored through the LIME algorithm. The results show that the established composite drought index combines the advantages of SPI and SBI in drought forecasting. The monthly and annual droughts in the JRB showed an increasing trend from 1991 to 2020, but the temporal characteristics of the changes in each subregion were different. The accuracies of the trained random forest model for heavy drought in Baizhiao (BZA) and Shaduan (SD) stations were 83% and 88%, respectively. Furthermore, the Local Interpretable Model-Agnostic Explanations (LIME) interpretation identified the essential precipitation, baseflow, and evapotranspiration features that affect drought. This study provides reliable and valid multivariate indicators for drought monitoring and can be applied to drought prediction in other regions.