We present $^{77}$Se-NMR measurements on FeSe$_{1-x}$S$_x$ samples with sulfur content $x=0,9,15$ and $29%$. Twinned nematic domains are observed in the NMR spectrum for all samples except $x=29%$. The NMR spin-lattice relaxation rate shows that magnetic fluctuations are initially enhanced between $x=0%$ and $x=9%$, but are strongly suppressed for higher $x$ values. The observed behavior of the magnetic fluctuations parallels the superconducting transition temperature $T_c$ in these materials, providing strong evidence for the primary importance of magnetic fluctuations for superconductivity, despite the presence of nematic quantum criticality in this system.
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