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