If you have the astringent types, such as 'Hachiya' that looks like an upside-down pear, you can pick them as soon as they turn deepest orange. They'll be hard but they ripen best off the tree. Once off the tree, allow them to turn almost-mushy soft to become edible-sweet without being astringent.

If you have the non-astringent type, such as 'Fuyu' that looks like a squat tomato, you pick it as soon as it's a nice color. It can be eaten while still firm. They will soften to mushy and can be used as the ones above but most people prefer to eat them firm in salads, etc..

Joe

Generally I would agree with Joe, except if you have birds, squirrels and raccoons, you may have to forego the waiting until the deepest orange color, and pick them now. Squirrels in particular can be nasty, and bite into every persimmon well before they are sweet/soft enough to eat. They do ripen well off the tree, and here in the East Bay, they are already colored up enough to pick.