We report 133 Cs NMR and 75 As Nuclear Quadrupole Resonance (NQR) measurements on the normal metallic state above Tc of a quasi-one-dimensional superconductor Cs2Cr3As3 (Tc < 1.6 K). From the 133 Cs NMR Knight shift 133 K measured at the Cs1 site, we show that the uniform spin susceptibility χspin increases from 295 K to ∼ 60 K, followed by a mild suppression; χspin then levels off below ∼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/T1T (the nuclear spin-lattice relaxation rate 1/T1 divided by temperature T ), shows an analogous trend as χspin. Comparison with the results of 1/T1T near Tc with K2Cr3As3 (Tc = 6.1 K) and Rb2Cr3As3 (Tc = 4.8 K) establishes a systematic trend that substitution of K + ions with larger alkali ions progressively suppresses Cr spin fluctuations together with Tc.