The structural and spectroscopic properties of the Be+-H-2 and Be+-D-2 electrostatic complexes are investigated theoretically. A three-dimensional ground-state potential energy surface is generated ab initio at the CCSD(T) level and used for calculating the lower rovibrational energy levels variationally. The minimum of the potential energy surface corresponds to a well depth of 3168 cm(-1), an intermolecular separation of 1.776 angstrom, with the bond of the H-2 subunit being 0.027 angstrom longer than for the free molecule. Taking vibrational zero point energy into account, the complexes containing para H-2 and ortho D-2 are predicted to have dissociation energies of 2678 and 2786 cm(-1), respectively. The nu(HH) band of Be+-H-2 is predicted to be red-shifted from the free dihydrogen transition by -323 cm(-1), whereas the corresponding shift for Be+-D-2 is predicted to be -229 cm(-1). The dissociation energy of the Be+-D-2 complex is calculated to be slightly higher than the energy required to vibrationally excite the D-2 subunit, raising the possibility that the onset of dissociation can be observed in the infrared predissociation spectrum at a particular rotational energy level in the nu(DD) manifold.