I guess there are many views over banana which contradict each other. In my parenting, I have grown up being fed that banana is a big 'no' if the baby is having cold and cough. But once my sister-in-law came to visit us from Ukraine and she said, "Banana is the best thing to give a baby if she has cold and cough, and doctors there advise parents to give banana." She may be right, but I dismissed the thought of implementing it on my baby, giving in to the fact that perhaps weather conditions too plays a role in it, and in the climate of Ukraine banana would work well for cough and cold but not here. So, I am not pretty sure as what to do, but personally I do not give my child banana if she has especially, cough. You can go through this link and get a hang of both the logical point of view as well. http://www.diethealthclub.com/askquestion/8056/banana-common-cold-is-it-true-that-if-you-eat-bana.html
There is a lot of confusion regarding this one, as some parents and doctors believe it is better to avoid bananas during cough and cold. Others feel bananas are good during cough and cold. The best way to decide is to first speak to your child's doctor. Also, you can try giving a banana once during cough and cold and see the effect. If you notice an improvement, you may continue, else stop giving the same. Here is a study based article you can go through to get some more clarity http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3326802/
A cold is a contagious viral infection, so obviously eating bananas cannot cause a cold. Well, bananas do increase mucus and phlegm build-up and this may cause additional respiratory distress. A 2010 study conducted by American scientists revealed that eating a slightly unripe Cavendish banana can actually reduce your chances of catching a common cold. A certain enzyme present in these bananas inhibits the cold-causing bacteria.