  Magnetic behavior of BaMoP$_2$O$_8$ with the spatially anisotropic triangular
arrangement of the $S=1$ Mo$^{4+}$ ions is explored using thermodynamic
measurements, neutron diffraction, and density-functional band-structure
calculations. A broad maximum in the magnetic susceptibility around 46\,K is
followed by the stripe antiferromagnetic order with the propagation vector
${\mathbf k}=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ formed below $T_N\simeq 21$
K. This stripe phase is triggered by a pronounced one-dimensionality of the
spin lattice, where one of the in-plane couplings, $J_2\simeq 4.6$ meV, is much
stronger than its $J_1\simeq 0.4$ meV counterpart, and stabilized by the weak
easy-axis anisotropy. The ordered moment of 1.42(9) $\mu_B$ at 1.5 K is
significantly lower than the spin-only moment of 2 $\mu_B$ due to a combined
effect of quantum fluctuations and spin-orbit coupling.
