In 2018 there are four ways to use your cryptowallets:
You should trust them because they can easily send your private keys to some servers: even if the code is opensourced, deployed one can have backdoors and exploits.
They are not very useful: you need to have your laptop to sign transactions. Moreover, you can't pay in cafe or vending just with these wallets.
There are 3 steps to pay in crypto:
We need internet to grap the data for transaction from blockchain and to broadcast the signed transaction. Private key is used only in the intermediate stage, which is just mathematical operations that don't require internet connection.