The only question that I had about water getting into the outlet had to do with potential damage to the EVSE plug or internal charger.
I thought I had phrased my answer to cover that as well as the personal safety question, but perhaps it wasn't clear enough.
There are two things that could damage the car:
Electrically speaking, there could be a path for current to flow between the two pins of the charger. This would require enough water (and water with some salt in it) in and around the charger to go from the deeply-recessed metal contact, out of the well it sits in, between the two wells, back into the other well, and down to the other deeply-recessed metal contact. It would
also require the signal pins to have not shorted out in the same way, which would prevent a charge from even starting. Even if all those conditions were met, the built-in safety on the charger should detect the overcurrent situation and shut down the charger almost immediately.
If all of the above were to happen/fail, it would heat up and melt the plastic in the handle or the car's port, but, basically, the chances thereof are effectively nil.
the other thing is much simpler: Water inside the connector managed to corrode the metal contacts. The contacts usually have corrosion-resistant plating (nickel, for example), to reduce the chance of corrosion of this sort, but it could theoretically happen. This is what I'd be much more cautious about personally, but the mitigation is just not letting much water get in the connector.
I previously hung the L1 charger that came with my Volt upside-down off of a faucet on the outside of the house over an eve, which kept it pretty dry, but I since bought one of these plastic "holster" things to attach to the house, which prevents any water (or bugs!) from getting in and looks a little nicer:
https://www.amazon.com/AmazingE-1001-0015-B-Charging-Wall-Mount-Connector/dp/B072MP3N1Y/
Just to add, for it to be a health hazard, a current path would have to exist between the internal metal contact (either the hot side on a L1 charger or either on an L2), to your hand, and your shoes would also need to have low enough resistance to the ground to form a current path. Basically, you and the charge handle would need to be covered with conductive water, the safeties would have to fail, and you'd need to still be touching it when the charge event started.
The second question, or actually first question since it was the OP's original question has to do with home outdoor EVSE installation and whether you need to keep the plug sheltered from weather while not in use.
I thought the answer to this has been more or less covered by suggestions to keep the plug relatively dry (and the charge door closed), but to state more explicitly, some water is fine but it's good to not let water sit inside it. My solution on the charger side was the above holster.
In a public charging situation the risk of unsafe equipment or installation is extremely low, but when using an outdoor outlet, or an indoor outlet with extension cords, or a DIY or semi-DIY outdoor installation, and possibly using the lowest cost EVSE found online with no research into company background, I would say maybe think twice about plugging in while it is raining, especially since we are talking about a PHEV where you can easily skip a charge session if needed.
The former is debatable (at work we had a L2 charger burn itself up internally due to poor electrical connections), but in any case, as mentioned by others the rest of the charger and the outlet you plug it in to is a completely different matter from the J1772 head.
Don't mess around with those. Period. You'll probably get away with it, but if you don't the cost is high.
A recently-built house is required to have GFI outlets in the garage, which helps from a life-safety standpoint, but even with that you definitely don't want to let the body of the charger or the outlet it's plugged into get wet at all. At best you could break the charger (I don't know that it's weather sealed) and at worst you could set your house on fire due to an overheated outlet. My solution was to plug the L1 charger the car came with into an internal outlet in my garage, string the longer charger cable under a gap beside the garage door, and plug in the car outside, but if you must plug in the charger to an external outlet put it in a location or enclosure in which it can't get wet (also make sure there's enough ventilation it doesn't overheat).
Also, make sure the outlet is in good condition, which it might well not be if you have an older house and it's a garage or outdoor outlet. After the car has been charging for an hour, feel the outlet and the plug and make sure it isn't too warm. If it's hot, you have poor connection and it could set your house on fire eventually. Replace the outlet with a new one--it literally costs like $2.