You concern is valid. Rabbits can and do injury themselves jumping from too high. I suspect that besides being able to get through the rails, there is probably also space between the steps (kick plate).
We have some barriers in our home that a bunny could get through so we use plastic lattice as in the image below. It is relatively inexpensive, and in our experience they don't chew on it.
If there are other pets in the house, that might scare them and make the bun try to take a short cut and jump to far, definitely do not leave spaces that they can fall long distances.
I also think it is possible to let the rabbit have access to the whole house without covering these spaces. If you do go that option, go slowly and supervise as they explore, if they do climb the stairs watch that they do so safely. If you have had your rabbit on your bed, you know how they might pace a bit and think about it before jumping. If you see that behavior, use a loud clap and strong NO to discourage them.
Rabbits don't like slippery surfaces. If the stairs are not carpetied they will likely not try them.
In general I would say that you should be able to make the entire house rabbit safe, and that she is going to live happier and longer with access to the whole house. If you have specific areas of concern, please post a new question with photos of the area so we can give you an answer specific to that.
Image from post https://pets.stackexchange.com/a/8130/13