This question in the OP is broader - or at least, I am choosing to interpret it more broadly: "when was the first date ever put on a coin?" Because different calendars were in use and used on coins long before the AD calendar was invented.
The earliest date on a coin is from the Hellenistic Greek series. Many Seleucid pieces bear dates in the Seleucid Era, a calendar which recognized Year 1 as the founding of the dynasty in 312 BC. The oldest date on a Seleucid coin that is recorded in Sear is Year 116 featured on a bronze coin of Antiochus III, which converts to 197-196 BC.
As for the question of why dates were put on coins in the first place: Dates on coins were, historically, part of the quality control system in an ancient or medieval mint. If a coin bears a date, then the government that issued it can know when the coin was made, and therefore how fine the metal is. They can also more easily detect if a mintmaster is issuing debased or underweight coins without approval. Putting dates on coins also makes it easier to recall coins of certain type or issue; you only have to instruct your people to keep an eye out for certain dates.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis