Where there is family, there is laundry. And for some reason, one person (you) seems to always end up having to deal with it. This system is simple enough so that eventually, any family member can handle it. (Unfortunately you DO have to wait till they can walk).
Buy one laundry basket for each family member and one for “house”. Permanently mark each basket with a person’s name, or use different color baskets. We made little signs and had them laminated at the copy shop, then simply “threaded” them in the basket’s holes. Now, when laundry comes out of the dryer, it gets folded and put in the correct basket. This can be a family chore or it can be done by the family’s “laundry expert”.
Once the clean clothes are in the baskets, the laundry expert washes her hands of them, so to speak. Each person is in charge of their basket. If your family is like ours, the baskets never leave the laundry room. Most people (who shall remain nameless) simply use their basket like a mini dresser so they never actually have to put anything away. But this matters not to you. If they want to go to the basement to change clothes, who cares? The laundry remains miraculously organized.