Transport properties of the classical antiferromagnetic XXZ model on the square lattice have been theoretically investigated, putting emphasis on how the occurrence of a phase transition is reflected in spin and thermal transports. As is well known, the anisotropy of the exchange interaction ∆ ≡ Jz/Jx plays a role to control the universality class of the transition of the model, i.e., either a second-order transition at TN into a magnetically ordered state or the Kosterlitz-Thouless (KT) transition at TKT , which respectively occur for the Ising-type (∆ > 1) and XY -type (∆ < 1) anisotropies, while for the isotropic Heisenberg case of ∆ = 1, a phase transition does not occur at any finite temperature. It is found by means of the hybrid Monte-Carlo and spin-dynamics simulations that the spin current probes the difference in the ordering properties, while the thermal current does not. For the XY -type anisotropy, the longitudinal spin-current conductivity σ s xx (= σ s yy ) exhibits a divergence at TKT of the exponential form, σ s xx ∝ exp B/ T /TKT − 1 with B = O(1), while for the Ising-type anisotropy, the temperature dependence of σ s xx is almost monotonic without showing a clear anomaly at TN and such a monotonic behavior is also the case in the Heisenberg-type spin system. The significant enhancement of σ s xx at TKT is found to be due to the exponential rapid growth of the spin-current-relaxation time toward TKT , which can be understood as a manifestation of the topological nature of a vortex whose lifetime is expected to get longer toward TKT . Possible experimental platforms for the spin-transport phenomena associated with the KT topological transition are discussed.