Charged kaon production has been measured in Si+Al and Si+Au collisions at 14.6 A GeV/c, and Au+Au collisions at 11.1 A GeV/c by Experiments 859 and 866 (the E--802 Collaboration) at the BNL AGS. Invariant transverse mass spectra and rapidity distributions for both K+ and K- are presented. The centrality dependence of rapidity-integrated kaon yields is studied. Strangeness enhancement is observed as an increase in the slope of the kaon yield with the total number of participants as well as the yield per participant. The enhancement starts with peripheral Si+Al and Si+Au collisions (relative to N+N) and appears to saturate for a moderate number of participating nucleons in Si+Au collisions. It is also observed to increase slowly with centrality in Au+Au collisions, to a level in the most central Au+Au collisions that is greater than that found in central Si+A collisions. The enhancement factor for $K^+$ production are 3.0 +-0.2(stat.) +-0.4(syst.) and 4.0 +-0.3(stat.) +-0.5(syst.), respectively, for the most central 7% Si+Au collisions and the most central 4% Au+Au collisions relative to N+N at the correponding beam energy.