Women should take the same precautions they would take anywhere else. Many women traveling to this region have to adjust to the prevalence of traditional chivalry. In the South, if a man opens a door for you, it’s considered a sign of respect, not condescension.
Another adjustment is the possible assumption that two or three women who go to a bar or tavern together might be there to invite male companionship. This misunderstanding can happen anywhere, but in some parts of the South it might be slightly more prevalent. Being aware of it is the best defense, otherwise no other steps need to be taken.