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We report $^{133}$Cs NMR and $^{75}$As Nuclear Quadrupole Resonance (NQR)measurements on the normal metallic state above $T_c$ of aquasi-one-dimensional superconductor Cs$_2$Cr$_3$As$_3$ ($T_c < 1.6$~K). Fromthe $^{133}$Cs NMR Knight shift $^{133}K$ measured at the Cs1 site, we showthat the uniform spin susceptibility $\chi_{spin}$ increases from 295~K to$\sim$ 60~K, followed by a mild suppression; $\chi_{spin}$ then levels offbelow $\sim$10~K. In contrast, a vanishingly small magnitude of $^{133}K$indicates that Cs2 sites contribute very little to electrical conduction andthe exchange interactions between 3d electrons at Cr sites. Low frequency Crspin dynamics, reflected on $^{75}$As $1/T_1T$ (the nuclear spin-latticerelaxation rate $1/T_1$ divided by temperature $T$), shows an analogous trendas $\chi_{spin}$. Comparison with the results of $1/T_1T$ near $T_c$ withRb$_2$Cr$_3$As$_3$ ($T_c=6.1$~K) and Rb$_2$Cr$_3$As$_3$ ($T_c=4.8$~K)establishes a systematic trend that substitution of K$^{+}$ ions with largeralkali ions progressively suppresses Cr spin fluctuations together with $T_c$.

$^{133}$Cs and $^{75}$As NMR investigation of the normal metallic state of the quasi-one-dimensional Cs$_2$Cr$_3$As$_3$