Background: In the late stages of stellar core collapse just prior to core bounce, electron captures on medium-heavy nuclei drive deleptonization. Therefore, simulations require the use of accurate reaction rates. Nuclei with neutron number near N=50 above atomic number Z=28 play an important role. Rates presently used in astrophysical simulations rely primarily on a relatively simple single-state approximation. In order to improve the accuracy of the astrophysical simulations, experimental data are needed to test the electron-capture rates and to guide the development of better theoretical models and astrophysical simulations. Purpose: The purpose of the present work was to measure the Gamow-Teller transition strength from Kr86 to Br86, to derive the stellar electron-capture rates based on the extracted strengths, and to compare the derived rates with rates based on shell-model and quasiparticle random-phase approximation (QRPA) Gamow-Teller strengths calculations, as well as the single-state approximation. An additional purpose was to test the impact of using improved electron-capture rates on the late evolution of core-collapse supernovae. Method: The Gamow-Teller strengths from Kr86 were extracted from the Kr86(t,He3+γ) charge-exchange reaction at 115MeV/u. The electron-capture rates were calculated as a function of stellar density and temperature. Besides the case of Kr86, the electron-capture rates based on the QRPA calculations were calculated for 78 additional isotopes near N=50 above Z=28. The impact of using these rates instead of those based on the single-state approximation is studied in a spherically symmetrical simulation of core collapse just prior to bounce. Results: The derived electron-capture rates on Kr86 from the experimental Gamow-Teller strength distribution are much smaller than the rates estimated based on the single-state approximation. Rates based on Gamow-Teller strengths estimated in shell-model and QRPA calculations are more accurate. The core-collapse supernova simulation with electron-capture rates based on the QRPA calculations indicate a significant reduction in the deleptonization during the collapse phase. Conclusions: It is important to utilize microscopic theoretical models that are tested by experimental data to constrain and estimate Gamow-Teller strengths and derived electron-capture rates for nuclei near N=50 that are inputs for astrophysical simulations of core-collapse supernovae and their multimessenger signals, such as the emission of neutrinos and gravitational waves.