We investigate the critical properties of the spin-1 honeycomb antiferromagnet BaNi₂V₂O₈, both below and above the ordering temperature T_N using neutron diffraction and muon spin rotation measurements. Our results characterize BaNi₂V₂O₈ as a two-dimensional (2D) antiferromagnet across the entire temperature range, displaying a series of crossovers from 2D XY to 2D XXZ and then to 2D Heisenberg behavior with increasing temperature. In particular, the extracted critical exponent of the order parameter reveals a narrow temperature regime close to T_N, in which the system behaves as a 2D XY antiferromagnet. Above T_N, evidence for Berezinskii-Kosterlitz-Thouless behavior driven by vortex excitations is obtained from the scaling of the correlation length. Our experimental results are in agreement with classical and quantum Monte Carlo simulations performed using microscopic magnetic model Hamiltonians for BaNi₂V₂O₈.