Magnetic behavior of BaMoP_2O_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 k=(1/2,1/2,1/2) formed below T_N≃ 21 K. This stripe phase is triggered by a pronounced one-dimensionality of the spin lattice, where

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... of the in-plane couplings, J_2≃ 4.6 meV, is much stronger than its J_1≃ 0.4 meV counterpart, and stabilized by the weak easy-axis anisotropy. The ordered moment of 1.42(9) μ_B at 1.5 K is significantly lower than the spin-only moment of 2 μ_B due to a combined effect of quantum fluctuations and spin-orbit coupling.