Self-explanatory quotes on the importance of open justice from senior judges in the developed world:
“Public scrutiny of the courts is an essential means by which we ensure that judges do justice according to law, and thereby secure public confidence in the courts and the law.” “Public awareness of what happens in our courts serves to bolster public confidence in the administration of justice.” - Lord Neuberger of Abbotsbury, Master of the Rolls for England & Wales.
“It is my personal belief that in any society which embraces the rule of law it is an essential requisite of the criminal justice system that it should be administered in public and subject to public scrutiny.” - Sir Igor Judge, Lord Chief Justice and President of the Courts of England and Wales.
“Justice should not only be done, but should appear to have been done.” - Lord Sankey in the Nottingham Journal Case  2 KB 1, 33.
Sound and very sacred part of the constitution
The open justice principle is "a sound and very sacred part of the constitution of the country and the administration of justice’. He went on to say, in rejecting the proposition that the courts could create new categories of exclusion: ‘to remit the maintenance of constitutional right to the region of judicial discretion is to shift the foundations of freedom from the rock to the sand." - Lord Shaw in Scott v Scott, House of Lords  AC 417, 473 “ . . . in the public trial is to be found, on the whole, the best security for the pure impartial and efficient administration of justice, the best means for winning for it public confidence and respect.” - Lord Atkinson in the same case.
"The fundamental rule is that judicial proceedings must be conducted in an open court to which the public and the press have access. A court cannot agree to sit in camera, even if that is by the consent of the parties." "One important manifestation of the principle [of open justice] is also the foundation of judicial accountability. I refer to the obligation to publish reasons for decision. This obligation requires publication to the public, not merely the provision of reasons to the parties.” - JJ Spigelman AC, former Chief Justice of New South Wales, Australia.
"The crucial prophylactic aspects of the administration of justice cannot function in the dark; no community catharsis can occur if justice is ‘done in a corner [or] in any covert manner’. It is not enough to say that results alone will satiate the natural community desire for ‘satisfaction’. A result considered untoward may undermine public confidence, and where the trial has been concealed from public view an unexpected outcome can cause a reaction that the system at best has failed and at worst has been corrupted. To work effectively, it is important that society’s criminal process ‘satisfy the appearance of justice’, and the appearance of justice can best be provided by allowing people to observe it." - Chief Justice Burger, in Richmond Newspapers v Virginia  USSC 154; 448 US 555 (1980). He went on to say that on occasion open justice must give way to “another overriding” interest : “Our holding here today does not mean that the First Amendment rights of the public and representatives of the press are absolute … [A] trial judge, in the interests of the fair administration of justice [may] impose reasonable limitations on access to a trial.” Quite rightly, the United States First Amendment jurisprudence, however, requires an extremely strong ‘overriding interest’ to displace the presumption of openness.
Putting all who participate in the trial under intense scrutiny
“A criminal trial is a public event. The principle of open justice puts, as has often been said, the judge and all who participate in the trial under intense scrutiny. The glare of contemporaneous publicity ensures that trials are properly conducted. It is a valuable check on the criminal process. Moreover, the public interest may be as much involved in the circumstances of a remarkable acquittal as in a surprising conviction. Informed public debate is necessary about all such matters. Full contemporaneous reporting of criminal trials in progress promotes public confidence in the administration of justice. It promotes the value of the rule of law." - Lord Steyn in Re S (A Child) (Identification: Restrictions on Publication)  UKHL 47;  1 AC 593,
"But a freedom which is restricted to what judges think to be responsible or in the public interest is not freedom. Freedom means the right to publish things which government and judges, however well motivated, think should not be published. It means the right to say things which ‘right-thinking people’ regard as dangerous or irresponsible. This freedom is subject only to clearly defined exceptions laid down by common law or statute." - Lord Hoffman in R v Central Independent Television plc  Fam 192, 202–3.
Peace, welfare and good government
“The general acceptance of judicial decisions, by citizens and by governments, which is essential for the peace, welfare and good government of the community, rests, not upon coercion, but upon public confidence.” - Chief Justice Gleeson, 'Public Confidence in the Judiciary' (2002) 76 Australian Law Journal 558.