Welcome to the site. Usually we see exterior drains seperate from interior but if your are tied together then you should not mix the two, driveway drain should not go into the house.
Do you have a perimeter dain around the outside of the bottom of the foundation?
Are you sure it connects inside the house?
If you have a pump in the basement, where does it pump to or, what is it connected to and where?
Short of any good answers a drain for the driveway with a sump and pump to pump away from the house or connect to same drain system but outside the house.