Not much - Brusilov stayed out of the conflict until 1920, and by that time the Whites had been driven back to the Crimea and Far East and were being finished off, and reaching them to join them would have bene both difficult and futile. It was at this point that he accepted an advisory position with the Red Army. Brusilov believed that, before any political priority, his mission was to save Russia from invaders and seperatists, and so it seems to me that he waited until one side had clearly won and was hence in a position to fend off the Poles, and then offered them his services.
If the Whites took Moscow in 1919, you might well see him throw in his lot with them; but that's an effect rather than a cause.