  We theoretically propose a mechanism for the anomalous Hall effect (AHE) in
an antiferrromagnetic (AFM) state of $\kappa$-type organic conductors. We
incorporate the spin-orbit coupling in the effective Hubbard model on the
$\kappa$-type lattice structure taking into account the orientation of the
molecules and their arrangement with dimerization. Treating this model by means
of the Hartree-Fock approximation and the linear response theory, we find that
an intrinsic contribution to the Hall conductivity becomes nonzero in the
electron-doped AFM metallic phase with a small canted ferromagnetic moment. We
show that, contrary to the conventional wisdom, the spin canting is irrelevant
to the Hall response; the nonzero Hall conductivity originates from the
collinear component of the AFM order in the presence of the spin-orbit
coupling. These features are well explained analytically in the limit of strong
dimerization on the anisotropic triangular lattice. Furthermore, we present an
intuitive picture for the present AHE by considering the real-space
configuration of emergent magnetic fluxes. We also find that the Hall response
appears even in the undoped AFM insulating phase at nonzero frequency as the
magneto-optical Kerr effect, which is enhanced around the charge transfer
excitations. We discuss possible detections of the AHE in ET based compounds.
