I don't have kids, but have travelled quite a bit. I have come across some fairly adventurous parents travelling with kids and staying in hostels. It is not the norm, but it is possible. If you really want to do it, I would start with something like 4 weeks in Costa Rica, and stay in places that are ~$20/night. That is long enough that you will have to do laundry, buy groceries, and learn to live as a family of backpackers. If the family can handle that, then you could go to more adventurous places.