We have used the Westerbork Synthesis Radio Telescope to search for H I 21 cm absorption associated with high redshift damped Lyα systems. We calculate limits to harmonic mean spin temperatures, <T_S_> for these systems, and find typical lower limits ~10^3^ K, consistent with previous results for other damped Lyα systems. We then analyze the relationship between the total HI column density and the integrated 21 cm absorption line equivalent width, EW_21_, for damped Lyα systems. We compare these results with similar results for lines of sight through our own galaxy. A trend is seen in which the damped Lyα systems show systematically lower EW_21_ values at a given total N(H I) than Galactic lines of sight, by a factor ~3-5. In the context of a two- temperature-phase model for the absorbing gas, these results suggest that for a given total neutral hydrogen column density, the damped Lyα systems contain a larger percentage of warm gas than is seen in typical Galactic lines of sight. We derive upper limits to H I 21 cm emission from the vicinity of the damped Lyα absorbers. The mass limits are adequate to rule out a substantial contribution by protoclusters of the type hypothesized by Oort [A&A, 139,211(1984)] to the observed statistics of damped Lyα systems at high redshift. Lastly, we derive a lower limit to the molecular hydrogen fraction in the cold gas phase for the absorption system towards 0528-2505 of: F= 2N(H_2_)/N(H I) > 2%.