The Gamow-Teller strength distribution from Sr88 was extracted from a (t,He3+γ) experiment at 115MeV/u to constrain estimates for the electron-capture rates on nuclei around N=50, between and including Ni78 and Sr88, which are important for the late evolution of core-collapse supernovae. The observed Gamow-Teller strength below an excitation energy of 8 MeV was consistent with zero and below 10 MeV amounted to 0.1±0.05. Except for a very-weak transition that could come from the 2.231-MeV 1+ state, no γ lines that could be associated with the decay of known 1+ states were identified. The derived electron-capture rate from the measured strength distribution is more than an order of magnitude smaller than rates based on the single-state approximation presently used in astrophysical simulations for most nuclei near N=50. Rates based on shell-model and quasiparticle random-phase approximation calculations that account for Pauli-blocking and core-polarization effects provide better estimates than the single-state approximation, although a relatively strong transition to the first 1+ state in Rb88 is not observed in the data. Pauli-unblocking effects due to high stellar temperatures could partially counter the low electron-capture rates. The new data serve as a zero-temperature benchmark for constraining models used to estimate such effects.