Based on three IPCC (Intergovernmental Panel on Climate Change) Representative Concentration Pathway (RCP) scenarios (RCP2.6, RCP4.5, and RCP8.5), observed meteorological data, ERA-40 reanalysis data, and five preferred GCM (general circulation model) outputs selected from 23 GCMs of CMIP5 (Phase 5 of the Coupled Model Intercomparison Project), climate change scenarios including daily precipitation, maximum air temperature, and minimum air temperature from 2021 to 2050 in the Heihe River basin, which is the second largest inland river basin in Northwest China, were generated by constructing a statistical downscaling model (SDSM). Results showed that the SDSM had a good prediction capacity for the air temperature in the Heihe River basin. During the calibration and validation periods from 1961 to 1990 and from 1991 to 2000, respectively, the coefficient of determination (R2) and the Nash–Sutcliffe efficiency coefficient (NSE) were both larger than 0.9, while the root mean square error (RMSE) was within 20%. However, the SDSM showed a relative lower simulation efficiency for precipitation, with R2 and NSE values of most meteorological stations reaching 0.5, except for stations located in the downstream desert areas. Compared with the baseline period (1976–2005), changes in the annual mean precipitation simulated by different GCMs during 2021–2050 showed great difference in the three RCP scenarios, fluctuating from −10 to +10%, which became much more significant at seasonal and monthly time scales, except for the consistent decreasing trend in summer and increasing trend in spring. However, the maximum and minimum air temperature exhibited a similar increasing tendency during 2021–2050 in all RCP scenarios, with a higher increase in maximum air temperature, which increased as the CO2 concentration of the RCP scenarios increased. The results could provide scientific reference for sustainable agricultural production and water resources management in arid inland areas subject to climate change.