It will depend on the facilities of the ship
, the stamina and heat tolerance of the child.
I have visited NY in Sept and I cruised the Greek Isles -early October.
It can still be very hot even in Sep - I assume that is not an issue??
European children will be back at school - so it will be quieter. All Europeans love children.
Another factor - what do you plan to do when you reach the islands - Greek Isles don't do sandy beaches like the Caribbean.
What cruiseline were you considering?? On my Oct Greek Isle cruise - I never saw any children.