Altermagnets offer a promising platform for dissipationless spin transport by combining zero net magnetization with spontaneous non-relativistic spin splitting. However, their magnonic transport properties remain largely unexplored. Here, we develop a quantum-kinetic theory for thermally driven magnon currents that cleanly separates Berry-curvature-driven intrinsic contributions from Drude-like scattering-dependent terms. Applying this framework to a collinear honeycomb antiferromagnet with anisotropic next-nearest-neighbor exchange and Dzyaloshinskii-Moriya interaction, we reveal spin-split magnon bands that support both intrinsic and extrinsic spin Nernst and Seebeck currents. For realistic parameters, we predict a sizable magnon spin-splitting angle (about 3.3 degrees) and a pure transverse spin current capable of exerting a strong spin-splitter torque suitable for magnetization switching.