I went in the month of fuly some years ago, and it was not that crowded. There is different tickets you can take. I took the one with the small visit of the onsen. Before going to the bath you can just chill in one of the room, where tea is served. As there is many bath, I even rember one where there was nobody.
Might depend on the day and on the season, but there is sure time when it's not crowded at all.
We went one evening in July last year. It was quite crowded. I wouldn't call the experience "relaxing" (I guess there are less famous, more relaxing onsen available in Japan), but it certainly was fascinating to see all the hustle and bustle behind the scenes. We had the full tickets, with our own private room for tea and snacks, plus a little tour through what are the emperor's bathing facilities, I believe. I would recommend a visit for the atmosphere, but not to relax.