Stressed nanomechanical resonators are known to have exceptionally high quality factors (Q) due to the dilution of intrinsic dissipation by stress. Typically, the amount of dissipation dilution and thus the resonator Q is limited by the high mode curvature region near the clamps. Here we study the effect of clamp geometry on the Q of nanobeams made of high-stress Si3N4. We find that tapering the beam near the clamps, thus locally increasing the stress, leads to an increased Q of MHz-frequency low order modes due to enhanced dissipation dilution. Contrary to recent studies of tethered-membrane resonators, we find that widening the clamps leads to a decreased Q despite increased stress in the beam bulk. The tapered-clamping approach has practical advantages compared to the recently developed "soft-clamping" technique, as it enhances the Q of the fundamental mode and can be implemented without increasing the device size.