  We investigate the critical properties of the spin-$1$ honeycomb
antiferromagnet BaNi$_2$V$_2$O$_8$, both below and above the ordering
temperature $T_N$ using neutron diffraction and muon spin rotation
measurements. Our results characterize BaNi$_2$V$_2$O$_8$ as a two-dimensional
(2D) antiferromagnet across the entire temperature range, displaying a series
of crossovers from 2D Ising-like to 2D XY 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
Berezinsky-Kosterlitz-Thouless behavior driven by vortex excitations is
obtained from the scaling of the correlation length. Our experimental results
are in accord with classical and quantum Monte Carlo simulations performed for
microscopic magnetic model Hamiltonians for BaNi$_2$V$_2$O$_8$.
