We have performed a pointed survey of N2D+ 2-1 and N2D+ 3-2 emission toward 64 N2H+-bright starless and protostellar cores in the Perseus molecular cloud using the Arizona Radio Observatory Submillimeter Telescope and Kitt Peak 12 m telescope. We find a mean deuterium fractionation in N2H+, R_D = N(N2D+)/N(N2H+), of 0.08, with a maximum R_D = 0.2. In detected sources, we find no significant difference in the deuterium fractionation between starless and protostellar cores, nor between cores in clustered or isolated environments. We compare the deuterium fraction in N2H+ with parameters linked to advanced core evolution. We only find significant correlations between the deuterium fraction and increased H_2 column density, as well as with increased central core density, for all cores. Towards protostellar sources, we additionally find a significant anti-correlation between R_D and bolometric temperature. We show that the Perseus cores are characterized by low CO depletion values relative to previous studies of star forming cores, similar to recent results in the Ophiuchus molecular cloud. We suggest that the low average CO depletion is the dominant mechanism that constrains the average deuterium fractionation in the Perseus cores to small values. While current equilibrium and dynamic chemical models are able to reproduce the range of deuterium fractionation values we find in Perseus, reproducing the scatter across the cores requires variation in parameters such as the ionization fraction or the ortho- to para-H_2 ratio across the cloud, or a range in core evolution timescales. ; Comment: Accepted to ApJ