We investigate the behavior of identical dipolar fermions with aligned dipole moments in two-dimensional multilayers at zero temperature. We consider density instabilities that are driven by the attractive part of the dipolar interaction and, for the case of bilayers, we elucidate the properties of the stripe phase recently predicted to exist in this interaction regime. When the number of layers is increased, we find that this "attractive" stripe phase exists for an increasingly larger range of dipole angles, and if the interlayer distance is sufficiently small, the stripe phase eventually spans the full range of angles, including the situation where the dipole moments are aligned perpendicular to the planes. However, in this regime, we expect the behavior to be strongly modified by the binding of dipoles between layers. In the limit of an infinite number of layers, we derive an analytic expression for the mean-field interlayer effects in the density-density response function and, using this result, we find that the stripe phase is replaced by a collapse of the dipolar system.