There is no harm in giving banana to your baby. It cannot cause cold or cough to your baby but any semi-solid food may be difficult for him in case the baby already has cold and congestion.
Rather, banana is one of the first few foods introduced to the baby after mothers milk. Fruits add to variety and taste. They can be given after mashing either alone or with cereals. Fruits like banana, mango, papaya, pear, plum and peaches are good sources of vitamin A and citrus fruits like oranges provide vitamin C besides minerals. Start with 1 to 2 teaspoonfuls and gradually increase the amount of food. Over 3 to 4 weeks increase to 50 to 60 g or half a cup (or one banana).