There is no written mention in the Torah, Halacha or Kabbala about tying a red string around the wrist. However, it seems to be a custom that has been around for some time, and may be based on Torah or Kabbalistic ideas. If there is any validity to the custom, it would be considered a "segula" or protective type of act.
A custom that is based on Torah ideas or mitzvoth may also have special segula properties on a smaller scale. Regarding the red string, the custom is to tie a long red thread around the burial site of Rachel, the wife of Jacob. Rachel selflessly agreed that her sister marry Jacob first to spare Leah shame and embarrassment. Later, Rachel willingly returned her soul to God on the lonely way to Beit Lechem, in order to pray there for the desperate Jews that would pass by on their way to exile and captivity. Often, one acquires the red string when giving charity.

Perhaps for these reasons the red thread is considered a protective segula. It recalls the great merit of our matriarch Rachel, reminding us to emulate her modest ways of consideration, compassion, and selflessness for the benefit of others, while simultaneously giving charity to the poor and needy. It follows that this internal reflection that inspires good deeds, more than the string itself, would protect one from evil and harm.