Drought risk has posed severe challenges to social development and human security in the Yellow River Basin (YRB), which is vulnerable to water scarcity. When dealing with drought, the insufficient coordination among various departments caused by the uncertainty of the comprehensive drought risk under climate change could lead to water conflict. Therefore, in this study, the drought risk changes in the YRB under shared socioeconomic pathway (SSP) scenario 5–8.5 were assessed by integrating the temporally dynamic hazard with the static vulnerability and exposure. Then, suggestions for drought risk management were developed according to the spatial differentiation of the drought risk characteristics. The results show that the drought hazard may increase as the standardized precipitation evapotranspiration index (SPEI) decreases from the baseline (1985–2014; −0.04 ± 0.26) to the mid-term period (2041–2070; −0.09 ± 0.21), and may decrease as the SPEI increases (0.14/10 yr) significantly (p < 0.01) during the long-term period (2071–2100). As a result, the drought risk in the upstream, midstream, and downstream areas may change by 0.26 ± 0.56, −0.05 ± 0.49, and −0.86 ± 0.38 levels in the long term, respectively, compared with the baseline. To reduce the drought risk, it is suggested that agricultural technology be further developed in the southwestern upstream, central upstream, and northern midstream areas; the water management be strengthened in the Hetao Plain and Ordos Plateau; the scale of exposure be controlled in the provincial capitals, southeastern midstream, and downstream areas; and the drought disaster research and early warning system be improved in the areas between 105°E and 110°E. Detailed suggestions for drought risk response were developed for each specific region.