The results of irradiations of isotopically enriched and natural Ge detectors with a neutron beam are presented. The analysis of the gamma-ray lines generated by decay of neutron induced unstable nuclei have provided us with a direct measurement of relevant interaction cross sections. Within a factor of 2, measurements and predictions are in good agreement. These results have important implications for the instrumental background in astrophysical gamma-ray spectrometers using germanium detectors. We confirm the reduction of the β-background component, which dominates the continuum background in the 0.1-1 MeV energy range, using 70Ge enriched detectors. We clearly identify β+ decays inside the detector as a significant source of continuum background in the 1-4 MeV energy range. This component is about 2 times more intense in 70Ge enriched detectors than in natural ones. This is mainly due to the enhanced yield of 69Ge and 68Ga isotopes. The choice of either natural or 70Ge enriched as optimum detector material depends on the energies of astrophysical interest. Detectors made of enriched 70Ge are more appropriate for studies at energies below 1 MeV. For higher energies natural germanium is slightly better. The possibility of rejecting most of β-background component by applying alternative analysis techniques makes natural Ge an appropriate material for future gamma-ray spectrometers.