The North American Nanohertz Observatory for Gravitational Waves (NANOGrav) has observed dozens of millisecond pulsars for over a decade. We have accrued a large collection of dispersion measure (DM) measurements sensitive to the total electron content between Earth and the pulsars at each observation. All lines of sight cross through the solar wind (SW), which produces correlated DM fluctuations in all pulsars. We develop and apply techniques for extracting the imprint of the SW from the full collection of DM measurements in the recently released NANOGrav 11 yr data set. We filter out long-timescale DM fluctuations attributable to structure in the interstellar medium and carry out a simultaneous analysis of all pulsars in our sample that can differentiate the correlated signature of the wind from signals unique to individual lines of sight. When treating the SW as spherically symmetric and constant in time, we find the electron number density at 1 au to be 7.9 ± 0.2 cm-3. We find our data to be insensitive to long-term variation in the density of the wind. We argue that our techniques paired with a high-cadence, low-radio-frequency observing campaign of near-ecliptic pulsars would be capable of mapping out large-scale latitudinal structure in the wind.