We have undertaken a survey of N₂H⁺ and N₂D⁺ toward 31 low-mass starless cores using the IRAM 30 m telescope. Our main objective has been to determine the abundance ratio of N₂D⁺ and N₂H⁺ toward the nuclei of these cores and thus to obtain estimates of the degree of deuterium enrichment, a symptom of advanced chemical evolution according to current models. We find that the N(N₂D⁺)/N(N₂H⁺) ratio is larger in more "centrally concentrated cores" with larger peak H₂ and N₂H⁺ column density than the sample mean. The deuterium enrichment in starless cores is presently ascribed to depletion of CO in the high density (>3 × 10⁴ cm^-3) core nucleus. To substantiate this picture, we compare our results with observations in dust emission at 1.2 mm and in two transitions of C¹⁸O. We find a good correlation between deuterium fractionation and N(C¹⁸O)/N(H₂)_{1.2 mm} for the nuclei of 14 starless cores. We thus identified a set of properties that characterize the most evolved, or "prestellar," starless cores. These are higher N₂H⁺ and N₂D⁺ column densities, higher N(N₂D⁺)/N(N₂H⁺), more pronounced CO depletion, broader N₂H⁺ lines with infall asymmetry, higher central H₂ column densities, and a more compact density profile than in the average core. We conclude that this combination of properties gives a reliable indication of the evolutionary state of the core. Seven cores in our sample (L1521F, Oph D, L429, L694, L183, L1544, and TMC 2) show the majority of these features and thus are believed to be closer to forming a protostar than are the other members of our sample. Finally, we note that the subsample of Taurus cores behaves more homogeneously than the total sample, an indication that the external environment could play an important role in the core evolution.