A screened lanai will always let water in. There is going to be wind with storms and the rain will be horizontal from the bottom of the screen all the way to the top, gutters won’t help. You will have to make it water tight between each aluminum post or it will leak.
The cheapest thing you can do is put weeping holes in the corners so any rain you get in your lanai will drain out or you will get a lot of standing water
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