Tourist traffic in RMNP does drop off quite a bit after Labor Day. You still get weekend visitors from Front Range towns, but it's far less crowded than the summer season. Especially during the middle of the week. The downside is there is more risk that weather will keep you from visiting some areas of the park. Trail Ridge Road is closed after the first big snow following Labor Day and that sometimes happens early in September.
Like most National Parks, if you do visit during high season, get out early in the day and stay late. Few park visitors get out early and most head back to town in the late afternoon. The popular spots can be almost empty before 9 AM or after 6 PM.
Other Colorado attractions, such as Mesa Verde close down some of their tours at the end of the summer.