You will get different advice from different people regarding which onsen areas are the best.

Also depends on what type of accommodation you're looking for (ex: Yugawara Onsen is good but lacks the variety of quality lodging that Kawaguchiko has), whether you're only interested in your hotel onsen or if you're looking for that quintessential onsen town feel (Kusatsu, Kinosaki, etc, would be good), your budget, accessibility, how much you value views, whether you want to stay on or get away from beaten path, whether you want modern or old-school, etc, etc.

However, most onsen aficionados will agree on the following as far as standards for a preferred onsen destination:
- Onsen town/area has healthy onsen output volume.
- Onsen town/area has gensen (the source hot spring water) that's of a respectable natural temperature (this is arbitrary, but you'd want at least >40C).
- Hotel/ryokan itself has its own gensen (jikagensen) with healthy output volume.

But it's also true that you can't just look at stats. For example, Kinugawa Onsen looks good on paper, but I'm not impressed by the quality of their onsen. In Hakone, the quality of onsen water varies from property to property, even if you're comparing two places that both advertise as having jikagensen with kakenagashi (= "free flowing", but this can be misleading, so I no longer put much stock in it) onsen. Ultimately you have to (or I do at least) look at the stats to narrow down the choices, then read reviews before choosing a place.

My favorite places, all things considered:
Kusatsu, Hakone (Miyanoshita, Gora, Sengahara area), Northern Alps area (Oku-Hotaka, Hirayu, etc), Noboribetsu.

I haven't been to many places in Kyushu... not a fan of Beppu (freecia's description is just about right), but Yufuin is good. Have always wanted to visit Kirishima Onsen, but it is not easy to get to. Bunch of great places in Nagano. Manza, Shirahone, Yudanaka, the list goes on. Some good places in Yamanashi also.