Snapshot VLBI observations at 5 GHz have been obtained for a subset of a sample of 30 quasars with extended radio structure. For all but one of the 12 sources involved, the objects were detected at one or more baselines. In 9 quasars, the visibility data revealed the presence of resolved core structure. Model-fitting indicated the nuclear emission of these sources to be quasi-linear and, in most cases, of core-jet type. Spearman rank correlations show that, for the total sample of 30 extended quasars, the projected radio size, the degree of jet curvature and the lobe arm length asymmetry are strongly correlated with the radio core luminosity, in the sense predicted by relativistic beaming models; they are uncorrelated with the relative core strength R, however. Follow-up observations of both pc-scale and overall radio morphologies of those quasars having resolved milliarcsec-scale structure are described in a companion paper