Well, I seem to have gained a reputation as the "onsen guy." Shhh! Don't tell anyone that we haven't onsened since January....
In any case, by popular demand I present to you my recommendations for family baths (In no particular order). These are places where you can either rent a separate family bath, or by virtue of their design, you get to hog the whole onsen until you are done.
If you need directions, see me, or buy one of those Zenrin map books (or see me to make copies of my Zenrin map book).
A great little ryokan in Kannawa with cool historical postcards and photos all throughout the lobby. There are 5 baths here, some indoor and some semi-outdoor. They are all "kashikiri."
It's run by a very charming woman named Ito-san, so be sure to say that I sent you!
Going up Odan Road, turn right just past the huge Marushoku. Pass Hyotan Onsen and it is on the right located on the corner of the next intersection. Turn right there to find the parking lot behind the inn.
Mugen no sato is located in Horita. From Route 11, turn at the Horita intersection and go down past Horita Onsen. Turn right at the little road just past the highway under pass. Go slow when it's dark because it is very easy to miss. Not more than 30 meters up the little road you take a left-hand turn, then immediately take another left onto the bridge that goes over the highway.
At the other end of the bridge is a 3 way intersection of some of the narrowest roads you will encounter in Beppu. Take the hard left. At night it looks like you are going to drive right off the cliff. Don't worry, you won't. You wind around a little rice paddy, then down a little hill and it widens out into a gravel road. By this time you can see the ethereal green and blue lights of the onsen. The onsen is very close to the highway, but you would never know. It is built straddling a river. Great scenery. And good for lightning bugs in summer.
* PS I didn't use the family bath here.
Mugen no Sato
Going up Odan Road, you will turn left on the little road at the Kannawa-Guchi highway bus stop. Tanimoto's parking lot is right behind the bus stop. The inn has 2 onsen that you "kashikiri" just by being in there. Perhaps weekdays are a better chance at getting a bath for a longer time. There is an indoor tub and an outdoor tub. Nice wooden interiors make for a great atmosphere.
I haven't used the family baths here, but there are several types to choose from. There is also a playground for the kids (Be sure they play before they bathe...) and a cool little knick-knack shop in the reception area.
Going up Odan Road, turn left at Oita Bank (Rt. 218). Just past the Kanpo no Yado and just before the Cosmo Drug Store, turn right up the hill. I think there is another turn up there somewhere but you can't miss the onsen. Plenty of parking, too.
Hyotan is a very famous and very large onsen. I have never used the private baths here, but if they are anything like the actual onsen then it must be good. The general manager, Mr Kouno, is one of the nicest guys you will ever meet with a nice thick Kansai accent. If you see him and his trademark moustache, be sure to say hi.
Going up Odan Road, turn right at the Marushoku. It's on the left with ample parking. You can't miss it.