A C-13 NMR study on kappa=(BEDT-TTF)(2)X (X = Cu(NCS)(2), Cu[N(CN)(2)]Br, and Cu[N(CN)(2)]Cl) is reported, where BEDT-TTF is bis(ethylenedithio) tetrathiafulvalene. The superconducting salts of X = Cu(NCS)2 and Cu[N(CN)(2)]Br show anamalous enhancement in nuclear spin-lattice relaxation rate T-1(-1) around 50 K, in spite of monotonic temperature dependence of the anisotropic Knight shift. On the other hand, the insulating salt of X = Cu[N(CN)(2)]Cl exhibits a divergent peak in (T1T)(-1) at 26-27 K, which is confirmed as a manifestation of an antiferromagnetic transition by broadening of the NMR line. Above 27 K, T-1(-1) shows a temperature dependence typical of critical magnetic fluctuations. The (T,T)(-1) and the line shape showed nearly the same profile for these three salts above 60 K, that gives experimental evidence for a correspondence between the anomalous enhancement of(T1T)(-1) in the superconducting salts and antiferromagnetic fluctuations established in the insulating salt of K type. The present results demonstrate the presence of strong electron correlation in both superconducting and insulating salts. Analytical expressions of T, and the Knight shift of the C-13 NMR for kappa-(BEDT-TTF)(2)X are given, which are general formulas applicable to other p pi electronic systems.