Using an e+e- annihilation data sample corresponding to an integrated luminosity of 3.19 fb-1 and collected at a center-of-mass energy s=4.178 GeV with the BESIII detector, we measure the absolute branching fractions B(Ds+→KS0K+)=(1.425±0.038stat.±0.031syst.)% and B(Ds+→KL0K+)=(1.485±0.039stat.±0.046syst.)%. The branching fraction of Ds+→KS0K+ is compatible with the world average and that of Ds+→KL0K+ is measured for the first time. We present the first measurement of the KS0-KL0 asymmetry in the decays Ds+→KS,L0K+, and R(Ds+→KS,L0K+)=B(Ds+→KS0K+)-B(Ds+→KL0K+)B(Ds+→KS0K+)+B(Ds+→KL0K+)=(-2.1±1.9stat.±1.6syst.)%. In addition, we measure the direct CP asymmetries ACP(Ds±→KS0K±)=(0.6±2.8stat.±0.6syst.)% and ACP(Ds±→KL0K±)=(-1.1±2.6stat.±0.6syst.)%.