The First Department, in a detailed decision, determined Supreme Court had properly balanced the requirement that a criminal trial be open to the public and the defendant’s right to a fair trial.  Because the court properly used its discretion to balance the two concerns neither mandamus nor prohibition was an available remedy.  The decision is worth reading—many of the issues discussed are not noted here:

The First Amendment guarantees the public and the press a qualified right of access to criminal trials … . This right must be kept in balance with the compelling interest of the defendant’s Sixth Amendment right to a fair trial and the right to privacy of prospective jurors … . The public’s right of access may be limited where there is a compelling governmental interest and closure is narrowly tailored to serve that interest … .

New York’s approach to courtroom closure is “comparable to the federal analysis” … . The press is not imbued with any special right of access, and while it possesses “the same right of access as the public,” it has no right to information about a trial that is “greater” or “superior” to that of the general public … . A ” trial judge has an affirmative constitutional duty to minimize the effects of prejudicial pretrial publicity'” … . Decisions to seal or disclose records fall within the inherent power of the court to control the records of its own proceedings … . While a court must guarantee that the defendant receives a fair trial, it must do so in a manner that balances the interests of “the defendant, jurors, witnesses, attorneys and the public at large” … . Matter of Daily News, L.P. v Wiley, 2015 NY Slip Op 02010, 3rd Dept 3-12-15