In addition to the direct action of impeaching some of the justices (as Joe W's answer explains) which is the best remedy against corrupt individuals, Congress also has the authority to do almost anything they want to the Court's makeup and procedures, if they can get a law to do so passed. So if it's more of a systematic flaw than just a few corrupt Justices, the system can be changed.
All the Constitution says about the makeup of SCOTUS is:
Article II: [The President] shall nominate, and by and with the Advice and Consent of the Senate, shall appoint, ... Judges of the supreme Court, ...
Article III: The judicial Power of the United States, shall be vested in one supreme Court, and in such inferior Courts as the Congress may from time to time ordain and establish. The Judges, both of the supreme and inferior Courts, shall hold their Offices during good Behaviour, and shall, at stated Times, receive for their Services, a Compensation, which shall not be diminished during their Continuance in Office.
The number of justices on the court and their policies and procedures are all set by law or custom, and can be changed by passing a new law. The only things off the table without an amendment are changing how Justices are appointed, removing them by means other than impeaching them, and docking their pay.
So Congress and the President working together can pack the court with "better" judges who aren't corrupt, redefine what constitutes a quorum, impose mandatory recusal guidelines (instead of the current self-recusal), change how cases get appealed to SCOTUS (but not that they can be), and so on.