We report on new measurements of Cabibbo-suppressed semileptonic D_{s}^{+} decays using 3.19 fb^{-1} of e^{+}e^{-} annihilation data sample collected at a center-of-mass energy of 4.178 GeV with the BESIII detector at the BEPCII collider. Our results include branching fractions B(D_{s}^{+}→K^{0}e^{+}ν_{e})=[3.25±0.38(stat)±0.16(syst)]×10^{-3} and B(D_{s}^{+}→K^{*0}e^{+}ν_{e})=[2.37±0.26(stat)±0.20(syst)]×10^{-3}, which are much improved relative to previous measurements, and the first measurements of the hadronic form-factor parameters for these decays. For D_{s}^{+}→K^{0}e^{+}ν_{e}, we obtain f_{+}(0)=0.720±0.084(stat)±0.013(syst), and for D_{s}^{+}→K^{*0}e^{+}ν_{e}, we find form-factor ratios r_{V}=V(0)/A_{1}(0)=1.67±0.34(stat)±0.16(syst) and r_{2}=A_{2}(0)/A_{1}(0)=0.77±0.28(stat)±0.07(syst).