What is an Independent Judiciary?

Independent of whom? Of what?

In India, many (almost and fourth) members of Parliament have criminal cases registered against them. 

Should the executive which is selected from amongst these have any say at all in the selection and promotion of judges? 

In some countries, like the United States (of America), judges to the highest court are appointed by the excutive, the president or governors. They are political appointees and the judgements of the courts there do reflect the political bias of their patrons.

Should India follow suit? 

You decide.

Here is an article in 'Deccan Herald' (Bangalore edition) datelined 21 Aug 2016 titled, "No judges, no justice" that is relevant to the issue.

Refer Link

http://www.deccanherald.com/content/565936/no-judges-no-justice.html

No judges, no justice


In the absence of any explanation from the Union government for not giving its assent to the 75 names recommended for appointment as high court judges, forwarded in February this year by the Chief Justice of India (CJI), one may presume that the Centre is not interested in processing the appointments quickly. Maybe, the government is unable to reconcile itself with the judgement striking down the National Judicial Appointments Commission (NJAC) Act and the 99th amendment to the Constitution last year. The government is yet to revise the Memorandum of Procedure (MoP) in consultation with the CJI and the Collegium. Media reports stated that the government wants to retain power to veto a recommendation on the grounds of national security.