Hawthorne Airport if cost isn't a factor.
Between LAX and BUR though. A person can consider which airport is easier to use and which airport is most conveniently located to get to and from the Staple Center. Which is more conveniently located can depend on time at which you will arrive at any airport. Right now at 8:16am, it is 40 minutes to drive the 17 miiles from Burbank to Staples Center. LAX is 37 minutes away for the similar miles. You can use Google maps to determine the usual travel times for the day and time you plan to travel.