When it comes to my passport, I guard it like I do my credit cards and social security number. When abroad or getting ready to travel internationally the only time I have my passport on my person is when I need to show it to airport personnel, or for border crossings. Even then, I keep it tucked inside a hidden pouch in my travel purse when not in use.
As soon as I get to the hotel where I’m staying, I lock my passport in the hotel safe. I also leave something ridiculous in the safe, like a left shoe, so that I don’t forget the passport is there. I might forget that my passport is in the safe, but I won’t pack just one shoe without looking for the other.
I do always carry a copy of my passport on my person while abroad so that I have proof of who I say I am if it’s needed. I’m not sure if my American driver’s license will suffice, so I keep the passport copy as a backup. I also keep a copy of my passport in all luggage so that if the luggage tag is missing I can prove that the luggage belongs to me.