Then everybody will use the old "currencies" like gold, sex and weapons. Actually after WW2 in Europe the situation was somewhat similar. Almost all means of communication and transportation was broken, supply chain didn't work and official currencies were worthless. Still people found ways to get food, medications, clothes and whatever else they needed by bartering their stuff or using gold jewellery as currency (e.g. one gold chain link = one potato). Others found other ways to do their shopping. Humans are innovative and highly adaptable.
BTW such a complete breakdown of the internet is nearly impossible. WW3 or a series really huge solar flares what keep hitting Earth can cause such a breakdown but in such scenarios money will be our least concern.