QUESTION: I am finishing off my basement and would like to add a service panel box in the basement. I have a 200amp panel box in my garage with 11 open slots out of 32. The builder pre wired a wire from the panel to the basement. The wire has a orange shield with #10 wire. There are 4 wires (3 and a ground). I would like to add a second service panel in the basement using the pre installed wire. My question is what size breaker should I use in the garage to go to my new service panel in the basement? I plan on adding a small service panel with 8 to 10 blanks. The will be for lights, outlets and baseboard heating units.

With the provided #10 wire, all you can connect is a 30 amp breaker, which may not be enough for what you plan on having in the basement, esp considering the electric heat. The 30 amp breaker/circuit may be just enough for the heat alone.
You will need to run a new larger cable, either #8 for 40 amp or #6 for a 50 amp subpanel, depending on the proposed load in the basement.