Spacial awareness is good to have no matter who you are or what your situation is. What's not good is being constantly worried or paranoid.
I look for lots of things. Who's in the area, what's in the area and what are different ways to get out if I need to. But I'm not specifically on the lookout for weirdos, as in, I'm constantly worried they'll be there or anything. If I notice someone sus I keep tabs on them until I don't think it's a threat.
One night I was out walking my dog, who was still under a year old (he was already a big dog though), and a guy is walking in the opposite direction. I see him coming but he's got a hoodie on and pulled up over his head, ducked in a way that I can't see his face. My radar didn't go off, but I made sure to watch him out of the corner of my eye as we passed each other. Then, he turns and heads straight for me.
So, I turned towards him and put my hands up. I couldn't outrun him with my dog, and I figured once I started making a commotion my dog might help out.
It turned out to be my (now-ex) fiance playing a trick on me, seeing what I would do. Once he saw me square up he threw back his hood and gave me a lecture on how I should have ran.Fucking asshole behavior, but I guess now I know what I'd do in that situation.
Anyway, no one's ever done that to me since, but I still keep my wits about me. I also don't think I'm constantly going to be followed by creepers either.