A ripe melon will develop a circular crack where the stem attaches to the fruit. Ripe melons will have a sweet aroma at the stem end. Ripe melons will slip easily off the stem; a half-ripe melon will require more pressure and may come off with half the stem attached. Leave melons on the vine until they are ripe.
For more information on growing Melons, check out this great guide from Harvest To Table: