If you want to be aware of your surroundings to protect yourself and be able to react to unexpected moments, yes, keep both of your eyes open when you're making pictures.
This is really helpful when you're shooting sports.
If you're locked on to a football receiver making catch, but another receiver is getting forced out of bounds towards you, your peripheral vision will kick in and sense that something is coming towards you.
It's also helpful when you're photographing an event like a reception or rally, where you may see a big moment unfolding out of the corner of your eye.
Here, my son uses his peripheral vision to eye chocolate chip cookies in the kitchen while posing with a camera he just painted!
