How is a malfunctioning heart valve repaired?

Depends: This is a very open-ended question. Dependent on the valve involved and the malady, the valve may be repaired or replaced. You have to be a bit more specific.

Open vs percutaneous: It can be done during open heart surgery using the by pass machine and keeping the heart at rest without beating. And in few cases by percutaneous approach, like in aortic stenosis with a contraindication for open heart. It can be replace or repair depending on the severity and condition.