You do not need to be connected to the Internet to make a transaction from your cold storage to your coinbase wallet account.
Generally, the answer will vary depending on the type of cold storage you are using.
if you have a hardware wallet, paper wallet or even using an open-source wallet, it is best to download electrum and then learn how you can send Bitcoin offline using that wallet "it supports hardware wallet & paper wallet."
It is better to do this using a clean computer (or fresh open-source OS,) which has not connected to the Internet before.