  The anomalous spin transport coefficients of gapped Dirac electrons are
studied with application to a quasi-two-dimensional organic conductor
$\alpha$-(BETS)$_2$I$_3$ in mind. In the presence of a gap induced by
spin-orbit interaction, we show that the effective Hamiltonian is similar to
the model considered by Kane and Mele with additional tilting. With this
effective Hamiltonian, conductivity tensors up to the linear order of the
applied magnetic field are obtained analytically using the microscopic linear
response theory or Kubo formula. It is shown that spin Hall conductivity and
anomalous diagonal spin conductivity proportional to the magnetic field become
nonzero in this system, which are written in terms of the Berry curvature and
orbital magnetic moment.The estimated values of spin conductivities using
typical parameters turn out to be comparable to the spin Hall conductivity in
Pt.
