Noise is random, but downsampling can reduce the randomness and thus the noise. Consider two flat frames taken with the Nikon D3 at ISO 3200. The standard deviation is the noise. If I measure the standard deviation of a 400 x 400 pixel area and compare it to the standard deviation of a 200 x 200 pixel downsize using bicubic in Photoshop, the standard deviation (the noise) decreases. This is confirmed by visual inspection of the images.
Indeed, downsampling will (by weighted averaging) cancel some of the highest frequency noise. But the important thing is that also the high spatial frequency (HSF) signal is reduced, often in quite a similar amount (so there is no real improvement). Of course the noise which is mostly random will be reduced and there will usually be some remaining signal at the new resolution, so there is some increase in S/N ratio at the expense of loss of detail.
Therefore the question becomes, do we measure image quality as lower noise but also having lost HSF signal, or as high Signal to Noise (with HSF detail to spare). I would favor the latter, which can be achieved with a dedicated noise reduction application quite effectively with only minimal reduction of HF signal. This keeps the larger output option intact, which is important to many.
If the OP's goal is to also reduce file size, and not only by lossless compression, then reducing noise before downsampling wil provide superior results. In the absense of noise the down-sampling filter becomes very important if aliasing artifacts are to be avoided.