  The fermionic part of the Schr\"odinger functional of QCD is formulated in
the lattice regularization with the staggered fermion. The boundary condition
imposed on the staggered fermion field are examined in terms of the
four-component Dirac spinor. The boundary terms are different from those of the
Symanzik's theory in the flavor structure due to the species doubling. It is
argued that, in the case of the homogeneous Dirichlet boundary condition,
surface divergence does not occur if the link variables of gauge field are
introduced on the original lattice, not on the blocked one. Its application to
the numerical calculation of the running coupling constant in QCD is discussed.
