Here are two ways in which the judicial branch influences the other branches:
Influence on the legislative branch:
• The Supreme Court has the power of judicial review, which allows it to declare laws and legislation unconstitutional. This places a check on the legislative branch and prevents it from passing any law it wishes. The legislative branch has to craft laws that will withstand potential judicial review.
Influence on the executive branch:
• The judicial branch can issue rulings and orders that the executive branch is obligated to enforce. For example, the Supreme Court can rule that a policy or action taken by the executive branch (President, agencies) is unconstitutional or improper. The executive branch has to comply with such rulings, even if it disagrees with them. This subjects the executive branch to oversight and limits on its discretionary powers.
So in summary, the judicial branch can exert meaningful constraints over the scope and application of powers of the legislative and executive branches through judicial review and issuance of binding rulings. This balance of powers across the branches is an important aspect of the separation of powers enshrined in the U.S. Constitution.