One of the building blocks of our democracy is that there are, in theory, three branches of our government that are supposed to operate independently of each other: the executive (President), legislative (House and Senate), and the judicial (US Supreme Court and other federal courts).  Senate Democrats have just pursued legislation to set a binding ethics code for the U.S. Supreme Court, following accusations by those legislators that some conservative justices had not disclosed some trips and real estate transactions. 
This would give Congress the power to hold hearings concerning justices of the Supreme Court.  As the court is known to be divided between the majority conservatives and the minority liberal justices, this would give Democrats the power to hold hearings and attempt the ouster of those conservative justices with whom they disagree.  
With the conservative majority in power at this time, liberal democrats have been debating how to change that balance of power.  Packing the court with additional justices has been proposed, although that has little chance of happening.  
Another avenue would be to seek to impeach existing conservative justices during the administration of a Democrat President.  If a conservative justice could be removed, the Democrat President could appoint a liberal justice in his/her place and change the balance of power.  
The proposed legislation could demand a justice recuse himself from and issue if said justice is seen as having a conflict of interest on an issue.  This is a very sticky point, as justices are generally appointed by Presidents who know their leanings on hot-button issues such as abortion.
I have reviewed the arguments in support of this legislation and have concluded this bill is pure, unadulterated politics.  It is an attempt by one branch of government to exercise control over another branch of government, and it may not even pass Constitutional muster.  
How much control should the House and Senate have over which justices decide cases before the Supreme Court?
None, I would argue.  Given the current political climate, I have no doubt such legislation would be used merely to whittle down the conservative majority and to allow the minority liberals on the Court to set policy for the nation by recusing conservatives as Congress chooses.
Ask yourselves:  Do you trust someone who is not in your party with having the power to exclude justices with whom they disagree to get rid of any justice that person chooses?   
I think not.
The Court needs to be free of threats by Congress and needs to maintain its independence.