Variations in the total solar irradiance (TSI) associated with solar activity have been argued to influence the Earth's climate system, in particular when solar activity deviates from the average for a substantial period. One such example is the 17th Century Maunder Minimum during which sunspot numbers were extremely low, as Earth experienced the Little Ice Age. Estimation of the TSI during that period has relied on extrapolations of correlations with sunspot numbers or even more indirectly with modulations of galactic cosmic rays. We argue that there is a minimum state of solar magnetic activity associated with a population of relatively small magnetic bipoles which persists even when sunspots are absent, and that consequently estimates of TSI for the Little Ice Age that are based on scalings with sunspot numbers are generally too low. The minimal solar activity, which measurements show to be frequently observable between active-region decay products regardless of the phase of the sunspot cycle, was approached globally after an unusually long lull in sunspot activity in 2008–2009. Therefore, the best estimate of magnetic activity, and presumably TSI, for the least-active Maunder Minimum phases appears to be provided by direct measurement in 2008–2009. The implied marginally significant decrease in TSI during the least active phases of the Maunder Minimum by 140 to 360 ppm relative to 1996 suggests that drivers other than TSI dominate Earth's long-term climate change.