Here we test the idea that new globular clusters (GCs) are formed in the same gaseous ('wet') mergers or interactions that give rise to the young stellar populations seen in the central regions of many early-type galaxies. We compare mean GC colors with the age of the central galaxy starburst. The red GC subpopulation reveals remarkably constant mean colors independent of galaxy age. A scenario in which the red GC subpopulation is a combination of old and new GCs (formed in the same event as the central galaxy starburst) cannot be ruled out, although this would require an age-metallicity relation for the newly formed GCs that is steeper than the Galactic relation. However, the data are also well described by a scenario in which most red GCs are old, and few, if any, are formed in recent gaseous mergers. This is consistent with the old ages inferred from some spectroscopic studies of GCs in external systems. The event that induced the central galaxy starburst may have therefore involved insufficient gas mass for significant GC formation. We term such gas-poor events 'damp' mergers.