Collisions between the winds of solar-like stars and the local interstellar medium result in a population of hot hydrogen gas surrounding these stars. Absorption from this hot H I can be detected in high-resolution Lyα spectra of these stars from the Hubble Space Telescope. The amount of absorption can be used as a diagnostic for the stellar mass-loss rate. We present new mass-loss rate measurements derived in this fashion for four stars (∊ Eri, 61 Cyg A, 36 Oph AB, and 40 Eri A). Combining these measurements with others, we study how mass loss varies with stellar activity. We find that for the solar-like GK dwarfs, the mass loss per unit surface area is correlated with X-ray surface flux. Fitting a power law to this relation yields M~F1.15+/-0.20X. The active M dwarf Proxima Cen and the very active RS CVn system λ And appear to be inconsistent with this relation. Since activity is known to decrease with age, the above power-law relation for solar-like stars suggests that mass loss decreases with time. We infer a power-law relation of M~t-2.00+/-0.52. This suggests that the solar wind may have been as much as 1000 times more massive in the distant past, which may have had important ramifications for the history of planetary atmospheres in our solar system, that of Mars in particular. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.