Wariness is the characteristic of being very cautious or careful. It's smart to have certain amount of wariness when you approach a dog you don't know.
Your wariness might cause you not to trust people when you first meet them — reserving judgment or waiting to see if they turn out to be reliable can both be evidence of wariness. Sometimes wariness is entirely appropriate, like when you move through a haunted house or handle a snake. The noun wariness comes from the adjective wary, or "cautious or careful," with its Old English root, wær, "prudent, aware, or alert."