Was it ethical for the Athenian courts to prosecute Socrates?

The Athenian political, social and legal system was not based on ethics - it was based of pragmatism and desired result. The Word we have borrowed from them today 'justice' to them meant 'settlement' - they were interested in a stable society free of internal dissent and vendettas, so they wanted disputes settled.

So the Word ethical was not part of their legal system. Athens wanted a settlement, and Socrates was sowing discord and dissent amongst the youth of his day - pushing oligarchy rather than democracy, and dissent from the way matters of state and society were progressed - he was regarded as a threat to national stability and internal security.

As a consequence he was charged with corrupting the youth and with impiety. The first one tells the problem, the second tells the solution (impiety carried the death sentence).

The court was not into ethics, but rather into solutions. It comprised 500 male citizens, and like all their courts, in a small population, these numbers were effectively the equivalent of a modern opinion poll - the court was an opinion poll of the citizens as a whole. A majority wanted him dead to protect their lifestyle, and that is how they voted. Ethics didn't cross their mind.

It is not valid to judge the Athenians by today's standards (and today's standards are questionable in their inconsistency and contradictions anyway). They had a community to run in a stable fashion, and the jurors made a judgement on what they wanted their society to be like and how to achieve it.

Incidentally, the jury was the court - there were no judges and lawyers to muddy the waters - just 500 citizens chosen by lot (randomly) representing the people and giving the verdict and sentence by secret ballot. They judged Socrates a threat to the stability and harmony of their country and wanted that removed.