  We investigate the effect of dimensional crossover in the ground state of the
antiferromagnetic spin-$1$ Heisenberg model on the anisotropic triangular
lattice that interpolates between the regime of weakly coupled Haldane chains
($J^{\prime}\! \!\ll\!\! J$) and the isotropic triangular lattice
($J^{\prime}\!\!=\!\!J$). We use the density-matrix renormalization group
(DMRG) and Schwinger boson theory performed at the Gaussian correction level
above the saddle-point solution. Our DMRG results show an abrupt transition
between decoupled spin chains and the spirally ordered regime at
$(J^{\prime}/J)_c\sim 0.42$, signaled by the sudden closing of the spin gap.
Coming from the magnetically ordered side, the computation of the spin
stiffness within Schwinger boson theory predicts the instability of the spiral
magnetic order toward a magnetically disordered phase with one-dimensional
features at $(J^{\prime}/J)_c \sim 0.43$. The agreement of these complementary
methods, along with the strong difference found between the intra- and the
interchain DMRG short spin-spin correlations; for sufficiently large values of
the interchain coupling, suggests that the interplay between the quantum
fluctuations and the dimensional crossover effects gives rise to the
one-dimensionalization phenomenon in this frustrated spin-$1$ Hamiltonian.
