Using 7.93 fb^-1 of e⁺e^- collision data collected at the center-of-mass energy of 3.773 GeV with the BESIII detector, we measure the absolute branching fractions of D⁰ → K^-e⁺ν_e, D⁰ → K^- μ⁺ν_μ, D⁺ → K<over bar>⁰e⁺ν_e, and D⁺ → K<over bar>⁰μ⁺ν_μ to be (3.521 ± 0.009_stat ± 0.016_syst)%, (3.419 ± 0.011_stat ± 0.016_syst)%, (8.864 ± 0.039_stat ± 0.082_syst)%, and (8.665 ± 0.046_stat ± 0.084_syst )%, respectively. By performing a simultaneous fit to the partial decay rates of these four decays, the product of the hadronic form factor f₊^K(0) and the modulus of the c → s Cabibbo-Kobayashi-Maskawa matrix element |V_cs| is determined to be f₊^K(0)|V_cs| = 0.7171 ± 0.0011_stat ± 0.0013_syst. Taking the value of |V_cs| = 0.97349 ± 0.00016 from the standard model global fit or that of f₊^K(0) = 0.7452 ± 0.0031 from the lattice quantum chromodynamics calculation as input, we derive the results f₊^K(0) = 0.7366 ± 0.0011_stat ± 0.0013_syst. and |V_cs| = 0.9623 ± 0.0015_stat ± 0.0017_syst ± 0.0040_LQCD.