The Barbour Collection covers vital records up to 1850; statewide recording of vital records (and subsequent indexes) starts on 1 July 1897. Between 1850 and 1897, however, you need to know the town where the birth, death or marriage occurred to find the record. These sources can help:
Connecticut Roots by Charter Oak Genealogy
Resources for Connecting with your Connecticut History and Heritage
Connecticut Roots by Charter Oak Genealogy 