No Grumps. No Chumps.

Some people are so starved for attention and validation that they feel they have to do things like this to get social media likes and clicks.

That one didn't seem like an attention getting stunt, from the video it looks like a clueless person thinking it's ok to step into a fountain to get some water. But as to the OP's question about what I would do; in this case, security was on the scene and so nothing needed to be done, but in other cases such as the clown that scratched his initials into the Colosseum wall; taking a video to hand over to authorities is a good idea, but more important would be saying something immediately while there's time to stop any more damage.

Call security immediately.

Stupid is a stupid does.

It's old news, that video was shown on Italian news channel La7 on August 2.

As you can see the local cop starts whistling when the woman is still inside the fountain, but she pretends not ot understand why. It's not clear if she was not fined or not.

Pretty soon they're going to rope off the Trevi Fountain and other sites for us to look at from a distance, like Stonehenge. Sad.

The danger here is that we all sit around and “tsk-tsk” about how stupid all these people are without realizing we are all just one step away from a stupid mistake ourselves. Maybe not as glaringly obvious as these, but still errors that impact the lives of others as we travel.

Stupid mistake or deliberate act? As a stupid mistake that is an excuse. All anyone has to do is look around and see that none else is walking in the fountain or defacing a monument. Doesn’t take much to figure out.
So, not a stupid mistake. More like the act of a stupid person, overall.