Hi Gus. Welcome to CT.
To anwser your question, Basil can't stand temperatures below about 50 degrees (F), so by putting them in a refidgerator that blows 43 degrees give or take is killing the plant. Basil is not a winter loving plant so if you arent going to use it quick enough, the best way to preserve basil is to dry it. It keeps just as much punch, looks great hanging in the kitchen, makes good gifts (dried herb bouqet) and will keep fairly well for weeks or even months.
Hope I anwserd your question!