Leave it on the vine.
Like almost all fruits, cantaloupes ripen only marginally after picking, and then only if they have been allowed to grow to full physical maturity on the vine. Unlike some fruits (apples, kiwis, pears, mangos, etc.), cantaloupes do not contain starches that will convert to sugar, so once they are cut from the mother vine, they will not become sweeter no matter what you do. Along with such fruits as peaches, nectarines and plums, however, cantaloupes can improve in color, texture and juiciness after being picked.
To aid this process, keep the melon in a loosely closed brown paper bag at room temperature. This will trap the ethylene gas that the cantaloupe releases and that helps it ripen. Putting a ripe banana or apple in the bag will generate more copious amount of the gas and speed the process. (Because the bag is porous, it allows carbon dioxide to escape and oxygen to enter, which is necessary if you don’t want the fruit to ferment.)
Since the first step in obtaining an edible, even delicious, cantaloupe is to buy one that is physically mature, look for ones that are well formed, heavy for their size, and with a skin color of tan or yellow, not green. Also, the netting on the skin should be raised; if it is flat, it’s a sign that the cantaloupe is immature. The stem end should be smooth and without ragged edges, indicating that the melon was mature enough to slip gently from the stem instead of suffering a wrenching premature separation.
If you smell the melon to get a sense of its sweetness and perfume, smell the opposite end - the blossom end - since that is where the softening and aroma begins.