Hi Dave,
You need to wait until they are ripe, bright red. The stems will bend over at that point since bugs cart away the berries. I'm not sure if any animals bother with them. But since they could be carted off by bugs, you can put organza bags over them to prevent that. If you don't have a female pack rat around who has a few organza bags stashed away, you can pick them up at a craft store. The link below shows what they are in case you have no clue what I'm talking about. I know my husband wouldn't, lol. : )

Here is a link that might be useful: Organza Bags

Remy, thanks. And you are right. I have absolutely no idea what organza bags are. Thanks for the link to the picture. I do have netting like that (for protecting flowers in hybridizing), so I can use it for the Jack-in-the-pulpit berries too.

But do you think when they are bright red, they are ready? Or do I have to wait until they dry up?

Hi again,
No you do not have to wait for them to dry up. If you are worried if they are ripe enough though, wait for them to start to flop. You have to remove all the moist berry flesh. Then since they need stratification either pick a spot in your garden and plant or do the winter sowing method.
Remy