No Balsa...

There are a couple of areas that need to be VERY STRONG as well as FREE from flex, and
I would like to suggest the following;

Imagine that you are looking at the boat from the rear towards the front at the location of the mast and keel, and for simplicity reasons here illustrated as a Triangle pointing down:
1/ The Keel attached in the bottom.
2/ The upper two corners is where the standing rigging that is supporting the mast is
... attached.
3/ The mast is standing on the center of the upper side (deck)

This area must be designed strong enough to be able to balance the involved forces, compression from the mast pushing down - the upward force on the standing rigging, the forces of the keel that actually work in all directions...

(A very simplified drawing is attached)

So back to your question;
One quite common design for the keel area is to build a box of a strong material that is attached to the hull...

I hope this simplified explanation makes sense to you, if not - just email me for further explanation.

Good luck