We report anisotropic spin polarization of edge currents in MoS2 and WS2 monolayer zigzag nanoribbons (ZNRs) deposited on either nonmagnetic or ferromagnetic insulator substrates. We employ an 11-band tight-binding model to calculate the electronic band structures of transition metal dichalcogenide (TMDC) monolayers and their corresponding nanoribbons in the presence of Rashba spin-orbit coupling (RSOC) and magnetic proximity effect produced by ferromagnetic substrate. We adopt the nonequilibrium Green's function method together with Landauer-Büttiker formalism to study the quantum transport behavior stemming from the edge states of ZNRs. We demonstrate that the spin-polarized edge current can be generated in both MoS2 and WS2 ZNRs with RSOC. We find that the spin polarization spreads out in all three directions. This is in stark contrast to what occurs in zigzag graphene nanoribbons, for which the polarization only exists in the transverse direction (across the width of ribbons). In addition, the spin polarization direction strongly depends on the strength of the intrinsic SOC component. The interplay of Rashba and intrinsic SOC is crucial for the spin polarization of the currents in any spatial direction. For TMDCs with stronger intrinsic SOC such as in WS2 monolayer ZNRs, we observe that the spin polarization along the perpendicular direction to the plane of the ZNR can be as large as 90%. Moreover, the unusual anisotropy of the spin polarization can be further enhanced by the magnetic proximity effect. These results open up possibilities for the generation of tunable high-spin polarization currents in ZNRs without application of an external magnetic field.