We report experimental studies of a series of BaFe$_2$S$_{3-x}$Se$_x$ ($0leq xleq3$) single crystals and powder specimens using x-ray diffraction, neutron diffraction, muon spin relaxation, and electrical transport measurements. A structural transformation from Cmcm (BaFe$_2$S$_3$) to Pnma (BaFe$_2$Se$_3$) was identified around $x = 0.7sim 1$. Neutron diffraction measurements on the samples with $x$ = 0.2, 0.4, and 0.7 reveal that the N${e}$el temperature of the stripe antiferromagnetic order is gradually suppressed from $sim$120 to 85 K, while the magnitude of the ordered Fe$^{2+}$ moments shows very little variation. Similarly, the block antiferromagnetic order in BaFe$_2$Se$_3$ remains robust for $1.5leq xleq3$ with negligible variation in the ordered moment and a slight decrease of the N${e}$el temperature from 250 K ($x=3$) to 225 K ($x=1.5$). The sample with $x=1$ near the Cmcm and Pnma border shows coexisting, two-dimensional, short-range stripe- and block-type antiferromagnetic correlations. The system remains insulating for all $x$, but the thermal activation gap shows an abrupt increase when traversing the boundary from the Cmcm stripe phase to the Pnma block phase. The results demonstrate that the crystal structure, magnetic order, and electronic properties are strongly coupled in the BaFe$_2$S$_{3-x}$Se$_x$ system.
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