Understanding how changing climatic conditions affect streamflow volume and timing is critical for effective water management. In the Rio Grande Basin of the southwest U.S., decreasing snowpack, increasing minimum temperatures, and decreasing streamflow have been observed in recent decades, but the effects of hydroclimatic changes on baseflow, or groundwater discharge to streams, have not been investigated. In this study, we determine how trends in precipitation, snowpack accumulation, and snowmelt rate relate to total streamflow, baseflow, and the hydrologic partitioning of baseflow and runoff at 12 sites in the Upper Rio Grande Basin (URGB) during 1980 to 2015. Total streamflow was partitioned into baseflow and runoff components at a daily time step using conductivity-mass-balance hydrograph separation. Trends in annual total streamflow, baseflow, runoff, baseflow index, precipitation, snowmelt rate, and peak snow water equivalent (SWE) were evaluated from 1980 to 2015 using the non-parametric Mann-Kendall trend test. Results indicate that baseflow forms a large component of total streamflow, contributing an average of 49% of total discharge upstream of Albuquerque, NM. During 1980 to 2015, decreasing trends in total streamflow occurred at 9 of 12 sites and were almost always associated with decreases in baseflow, suggesting that baseflow volumes can respond to changing climatic and anthropogenic conditions within decades. Decreasing snowmelt rates were more frequently associated with decreases in baseflow and total streamflow than were decreases in precipitation and peak SWE, highlighting the importance of snowmelt rate as a process controlling streamflow generation. If snow accumulation and snowmelt rates continue to decrease in the future, results indicate that total streamflow and baseflow volumes will decline, and that baseflow will become a larger fraction of total streamflow in the URGB.