A search for neutral hydrogen 21 cm emission has been made in the directions of the globular clusters 47 Tuc, NGC 1851, ω Cen, and NGC 6388, using the Parkes 64 m telescope. These clusters were chosen for observation because they have relatively large central escape velocities. Theoretical calculations indicate that such clusters are the most likely to accumulate the material which is lost from cluster stars. In the cases of NGC 6388 and 47 Tuc, emission from the Galactic plane is observed close to the systemic velocity of the clusters, making it difficult to identify any weak features which might be associated with the clusters themselves. A significant 21 cm emission profile is observed in the direction of ω Cen, blueshifted by ∼40 km s-1 with respect to the cluster. It is likely however that this emission arises from a high-velocity cloud that has a hydrogen column density of NH ∼ 3 × 1018 atoms cm-2. This cloud is probably part of a complex of high-velocity clouds that form the northern extension of the Magellanic Stream near Galactic longitude l ∼ 300°. No emission is observed to be present at the radial velocity of NGC 1851, and an upper limit of ∼9 M⊙ can be placed on the amount of neutral hydrogen within this cluster. The observational upper limits on the amount of interstellar hydrogen, both neutral and ionized, that can be present in NGC 1851 are a factor of ∼2.5 smaller than the gas mass which is predicted to have accumulated in the cluster core as the result of stellar mass loss. The lack of gas within globular clusters can be explained by assuming that most of the mass loss from stars on their first evolution up the giant branch takes place via a relatively fast wind (velocities ≳80 km s-1). Observed red giant winds are neither fast enough to escape a globular cluster nor substantial enough to account for the ∼0.2 M⊙ of material which must be lost in order to explain the morphology of the horizontal branch in globular cluster color-magnitude diagrams. It is suggested that the high-velocity mass loss required to explain the gas deficiency of globular clusters is driven by the helium core flash within red giants.