what rule determines whether the person inbounding the ball can run the baseline?

Welcome.
 

Ugafan,
This is allowable after every made basket.
mick

I will try to explain it.
 

What determines it is whether the throw-in is a result of a made basket. If a basket is made, a team cannot lose that priviledge until the throw-in is over. Or they cannot lose the entire baseline if a foul or violation (by the opponent) has occured before the throw-in has ended, and the ball will be taken back to the baseline.

I hope that clears it up.

Peace

NFHS 7-5-7
 

... After a a goal or a goal awarded as in 7-4-3 (successful free throws for fouls), the team not credited with the score shall make the throw-in from the end of the court where the goal was made and from any point outside the end line. A team retains this priviledge if the scoring team commits a violation or a foul (before the bonus is in effect) and the ensuing throw-in spot would be on the endline. Any player of the team may make a direct throw-in OR s/he may pass the ball along the end line to a teammate(s) outside the boundary.

Re: Welcome.
 

Awarded baskets (goaltending, BI), too.