In such a situation, if you've chosen to use an electronic cryptocurrency dependent on things like robust network and electrical infrastructure to function as your primary value store, you're going to have a bad time.
This doesn't even contemplate the additional issue for bitcoin and its derivatives: the blockchain depends on a consistent, global view of network transactions, and apocalyptic-type network segmentation will cause severe forking in the blockchain, further diminishing the utility and therefore the value of the [already worthless] cryptocoins.