Details on anisotropic magnon dispersion in van der Waals (vdW) ferromagnetic insulators CrPS4 and CrSBr are reported, driven by anisotropic Heisenberg exchange couplings arising from in-plane broken crystal symmetry. The anisotropic magnon dispersion contributes to longitudinal and transverse magnon currents generating the anisotropic spin Seebeck effect (ASSE) and the thermal Hall effect (THE) accompanied with spin Nernst effect (SNE), requiring neither external magnetic field nor Berry curvature. In CrPS4, the ASSE exhibits a very large anisotropy ratio of over 100% as the thermal gradient along different main axes, and this ratio can be further tuned by temperature or a gate current. The THE and SNE unconstrained by spin-orbit coupling (SOC) emerge when the thermal gradient is not parallel to the main axis, characterized by a large Hall angle approximate to 0.4. Compared to CrPS4, CrSBr exhibits a more limited anisotropic magnon transport owing to the less variation in magnon group velocities along different main axes. Moreover, the reversed magnitude relationship of magnon group velocities leads to the transverse magnon current being oriented in the opposite direction. These findings identify low-symmetry vdW magnetic materials as a promising framework for generation and manipulation of anisotropic magnon transport, relevant for spincaloritronic devices in the ultrathin regime.