Ransoming captured enemy warriors was indeed reasonably common in ancient Greece. When a battle line collapsed, those who could see no hope in running might surrender, leaving the victor with many prisoners:
And they clashed with the Boiotians and won a great victory, and they killed many, and took 700 of them prisoner.
-- Herodotos 5.77.2
There was a more or less standard price that a prisoner would fetch if ransomed. Around 500 BC, this price was 200 drachmai. A hundred years later, the typical seems to have dropped to 100 drachmai. Even so, this was slightly more than the amount you could get for the same prisoner by selling him as a slave (the typical alternative), which made ransoming an attractive option.
However, it should be clear from the existence of a "set rate" that ransoming in Classical Greece was quite different from the practice of Medieval Europeans. There was no point in trying to pick the richest men to take prisoner, and there doesn't seem to have been any negotiation to establish the proper price for individual captives. It wasn't about capturing certain people for maximum profit, but rather about selling them all back in bulk.
As a result of this, there is no evidence of any Greeks making an effort to capture rather than kill. While there was some money to be made through ransoming, it wasn't enough to inform the Greek approach to combat. Indeed, there is much more evidence of Greeks killing those who are already at their mercy. Consider this scene from the Iliad:
Adrastos clasped him by the knees and begged him: “Take me alive, son of Atreus, and accept a worthy ransom; many treasures lie stored in the palace of my wealthy father, bronze and gold and hard-worked iron; my father would give you ransom past counting, should he hear that I am alive at the ships of the Achaians.”
So he spoke, and sought to persuade the other's heart in his breast, and indeed, Menelaus was about to give him to his charioteer to lead to the swift ships of the Achaians, but Agamemnon came running to meet him, and scolded him, saying: “Soft-hearted Menelaus, why do you care for the men like this? Has so great kindness been done to you in your house by Trojans? Let not one of them escape sheer destruction and the might of our hands, not the boy whom his mother bears in her womb; let not even him escape, but let all perish together out of Ilios, unmourned and unmarked.”
So spoke the warrior, and turned his brother's mind, for he spoke wisely; so Menelaus with his hand thrust from him the warrior Adrastos, and lord Agamemnon struck him in the side, and he fell backward; and the son of Atreus planted his heel on his chest, and drew out the ashen spear.
-- Iliad 6.45-67
In the Classical period, death was still a common fate for those who were rendered helpless in battle. After one side broke, the other side's efforts were not marked by a desire to capture as many as possible, but to kill the fleeing enemy with impunity:
For when cities defeat their opponents in battle, words fail to express the joy they feel in the turning of the enemy, in the pursuit, in the killing of the enemy – such pride they feel in the work! Such shining glory they gain, such happiness at the thought of having enhanced their city! Everyone claims that they had a share in the plan, that they killed the most…
-- Xenophon, Hieron 2.15-16
There doesn't even seem to be any awareness here of the fact that there was money to be made by ransoming captives. The aim was to kill and destroy the enemy; captives were a burden; ransoming was an easy way to lose that burden, but so was murder.
Sources:
Peter Krentz, 'Fighting by the Rules: the Invention of the Hoplite Agon', Hesperia 71 (2002) 23-39
Hans van Wees, Greek Warfare: Myths and Realities (2004)
John Dayton, The Athletes of War: An Evaluation of the Agonistic Elements of Greek Warfare (2005)
Hans van Wees, 'Defeat and Destruction: the Ethics of Ancient Greek Warfare', in Tausend, S. (ed.), “Böser Krieg”: Exzessive Gewalt in der antiken Kriegsführung und Strategien zu deren Vermeidung (2011), 69-110
Yes, ransoming soldiers was important in Greek warfare, though this wasn't the goal of hoplite warfare. Most battles were big pushes where one army just tried to push the army off the battlefield and send them into retreat. That does not mean that ransom was not important though. Prisoners of war could be ransomed back to their city or sold as slaves to make money for the polis.
In the Peloponnesian War there are a couple of major examples. The first is the Battle of Sphacteria where the Athenians captured a few hundred Spartan soldiers (there were very few true Spartan soldiers so this was a major achievement). The Athenians used these prisoners as their main bargaining tool to form the Peace of Nicias and end the first phase of the war.
Another big example is the Siege of Syracuse. After the Athenian siege was routed, the Athenians were harried by the Syracusan cavalry trying to retreat to allies on Sicily and eventually encircled and sold into slavery. The loss of manpower likely cost the Athenians the war.