There is accumulating observational evidence suggesting that damped Lyalpha absorption systems are the progenitors of present-day spiral galaxies. We use the observed properties of these systems to place constraints on the history of star formation in galactic disks, and on cosmological theories of structure formation in the universe. We show that the observed increase in OMEGA(HI) Contributed by damped Lyalpha systems at high redshift implies that star formation must have been considerably less efficient in the past. We construct a model in which gas is converted into stars with an efficiency that increases with time, and we show that this model can reproduce most of the observed properties of damped Lyalpha systems, including the observed distribution of column densities. We also show that the data can constrain cosmological models in which structure forms at late epochs. A mixed dark matter (MDM) model with OMEGA(nu) = 0.3 is unable to reproduce the mass densities of cold gas seen at high redshift, even in the absence of any star formation. We show that at redshifts greater than 3, this model predicts that the total baryonic mass contained in dark matter halos with circular velocities V(c) > 35 km s-1 is less than the observed mass of H I in damped systems. At these redshifts, the photoionizing background would prevent gas from dissipating and collapsing to form high column density systems in halos smaller than 35 km s-1.