It is still unclear why the transition temperature T c of cuprate superconductors falls with underdoping. The doping dependence of the critical magnetic field H _c2 is directly relevant to this question, but different estimates of H _c2 are in sharp contradiction. We resolve this contradiction by tracking the characteristic field scale of superconducting fluctuations as a function of doping, via measurements of the Nernst effect in La _1.8-x Eu _0.2 Sr _x CuO ₄ . H _c2 is found to fall with underdoping, with a minimum where stripe order is strong. The same non-monotonic behaviour is observed in the archetypal cuprate superconductor YBa ₂ Cu ₃ O _y . We conclude that competing states such as stripe order weaken superconductivity and cause both H _c2 and T _c to fall as cuprates become underdoped.