Banking has existed for centuries, dating back to when most people were peasants. When did it become common for "regular" people to have their own personal bank accounts?

While it is hard to say what was or was not common, the evidence of Roman banking, such as the Sulpicii Archives, shows that it was an activity participated in by many different segments of society. In fact Jean Andreau's Banking in the Roman World argues that bamking as such was not participated in by the upper classes, who instead relied on elite financiers (like Cicero's friend Atticus) who were known personally to them. It is probably a somewhat overstated argument, but not by too much.

This doesn't mean that every peasant had a bank account, far from it, but money loans and deposits were available to even the not well off.