Experience shows that onion keep well at room temperature. In fact, they actually last longer, which means that you will waste less food. Onions are also easier to cut, slice and dice when they are at room temperature, which will reduce your preparation time.
However, once they are cut, sliced or diced, introducing them to the refrigerator is a good idea. Of course, after cutting, unused onions should be sealed in a plastic bag or container before being placed in the vegetable drawer.