The only real way to measure the perceived legitimacy of courts such as the Supreme Court is through opinion polling.
Legitimacy, when applied to courts, is defined as the belief that the court has the right to hold the power that it holds. In our system, there is a problem with the legitimacy of courts like the Supreme Court of the US. These courts are staffed by unelected judges who are not answerable to anyone. This makes it very possible for it to seem that they have no legitimacy in a system that is otherwise democratic.
The only way to determine whether people believe that the Court has legitimacy is to ask them. Scholars (as in this link) have to ask people various questions to determine whether the people feel the courts are legitimate. They might ask, for example, if they feel the Court generally bases its decisions on the Constitution. Or they might ask if people believe that Congress should be allowed to overrule Court decisions.
Because legitimacy is a matter of popular opinion, the only way to measure it is by asking about people's opinions.