  The anomaly of a discrete symmetry is defined as the Jacobian of the
path-integral measure. Assuming that the anomaly at low energies is cancelled
by the Green-Schwarz (GS) mechanism at a fundamental scale, we investigate
possible Kac-Moody levels for anomalous discrete family symmetries. As the
first example, we consider discrete abelian baryon number and lepton number
symmetries in the minimal supersymmetric standard model with the see-saw
mechanism, and we find that the ordinary unification of gauge couplings is
inconsistent with the GS conditions, indicating the possible existence of
further Higgs doublets. We consider various recently proposed supersymmetric
models with a non-abelian discrete family symmetry. In a supersymmetric example
with $Q_{6}$ family symmetry, the GS conditions are such that the gauge
coupling unification appears close to the Planck scale.
