I really cannot see why the Brownian motion is linked with the being of atoms. Cannot it be explained without atoms? You just need some fluctuations.
More precisely it has to do with existence of molecules. The demonstration was obtained after Einstein and Smoluchowski developed a QUANTITATIVE theory (based on the molecular theory) whose predictions could be measured. Measuring parameters of Brownian motion (by Perrin) confirmed the theory. There was no better (or competing) explanation of these measured parameters.