Experiments on underdoped cuprate superconductors suggest an intricate relation between the normal-state Nernst effect and stripe order. The Nernst signal appears enhanced near 1/8 hole doping and its onset temperature scales with the stripe-ordering temperature over some range of doping. Here, we employ a phenomenological quasiparticle model to calculate the normal-state Nernst signal in the presence of stripe order. We find that Fermi pockets caused by translational symmetry breaking lead to a strongly enhanced Nernst signal, with a sign depending on the modulation period of the ordered state and other details of the Fermi surface. This implies differences between antiferromagnetic and charge-only stripes. We also analyze the anisotropy of the Nernst signal and compare our findings with recent data from La1.6-x Nd0.4 Srx CuO4 and YBa2 Cu3 Oy.