It varies a lot. The brown color is caused by developer oxidation products, which accumulate as the developer becomes exhausted. But the degree of exhaustion versus the color is not regular and can not be used as a reliable indication. Some developers become exhausted before any brown appears. But the original Agfa Rodinal stock solution was dark brown, and it lasted almost forever in its sealed little vials. Amidol developers, which last only a few hours anyway, usually become quite brown about half way through their useful life. Dektol is useless when it becomes yellow. Anything beyond a light tan usually means HC-110 stock is no longer reliable.
Developer exhaustion usually happens by oxidation of the developing agents. That happens in a lot of ways. First is the oxidation that occurs when developing - the normal developing action. Next comes oxidation from air in contact with the solution, either in the bottle or in a tray or tank. Then there is oxidation (and other chemical reactions) from contamination. This is subtle. The chemicals used to make solutions are not absolutely pure, and some of the impurities may affect the developer (either slowly or rapidly). The same thing goes for the water. Contaminants, both chemical and bacterial, can also enter from the air. Except for exhaustion through use, these are usually unpredictable for the small darkroom.
You really want your darkroom solutions to be reliable so that they give repeatable results - so you can follow the same routine and get comparable results every time. Developers do not die all at once. As they become exhausted, they first begin to act more slowly. This often happens long before any color change appears. As they become weaker, they also develop photographic materials differently, changing the curve and often increasing fog and maybe stains. You want to avoid this for consistent results. Large laboratories use replenishers to compensate; most small users simply replace their solutions often. Most manufacturers list capacities and working lives for their products. Pay strict attention to these; treat them as absolute maxima and generally replace solutions well before their end of life. One shot developers for film help a lot, but pay attention to the shelf life of the opened stock solution.