Really has nothing to do with not having a storm door.
As we have seen hundreds of times someone installed a deck, stoop, porch whatever that is outside the door to close to the threshold.
Moden codes call for it being no closer then 4".
It's 100% sure way to get water to come in.
Also I'd bet the door was installed wrong in the first place.
Should have been done something like this.
It can be made from coil stock, peel and stick window tape or a store bought one like this.
No amount of caulking on the outside is going to stop it.
The whole door would need to be removed to do this right and fix any damage under it.