We use SSB (Ham) radios now. Neither offshore recreational or commercial maritime traffic would be effected if the satellites or Internet went down. That would effect GPS and navigational systems, but it would have absolutely NO effect on communications via radio.
CB radios much like marine VHF radios have very limited distance (right conditions, maybe 16 miles on a commercial vessel, and maybe only 3 miles or so, on a recreational vessel). These radios work on line of sight or "ground waves".
SSB however is a different story. When you listen to a local AM broadcast station, you are receiving the ground wave signal. The ground wave travels directly along and near the ground for often a hundred miles or so from the transmitter location.
At sea, a good SSB can be 16 times more powerful then an AM broadcast station. Sky waves travel toward the sky, rather than hang out along the ground. You would not be able to hear the sky-wave signals, except for the ionosphere. The ionosphere is many miles above the earth, Here, the ionosphere is bombarded by radio sky waves, x-rays, ultraviolet rays, and other forms of high-frequency waves.
When a sky-wave signal reaches the ionosphere, it will refract the signal, bending it back to earth. The signal can then be heard where the signal reaches the earth, but depending on a number of variables, there might be an area where no signal from that particular transmitter is heard. This area is the skip zone, but after the sky-wave signal bounces on the earth, it will return toward the sky again, and therefore the signal goes skipping around the world.
Now, lets add a bit of updated technology and newer SSB radio. Suppose you have a typical NEW 5-kW SSB broadcast transmitter. You will only be able to impress 2.5 kW of audio power on that signal. This means that each of the two sidebands will have only 1.25 kW of power. But in the newer highly effective communications using SSB, a single sideband signal removes the carrier and one sideband and concentrates all of its energy in one sideband. Thus, a 1-kW SSB signal will "talk" as far as a 4-kW conventional AM or FM transmitter. It is one reason why long distances can be covered effectively with SSB. These newer SSB radios have the efficiency of an SSB signal that is 16 times greater than with a conventional-commercial AM radio broadcast signal.
Furthermore, there are ship to shore and relay SSB & Ham stations all around the world. Never in all my experience at sea, have I ever not been able to listen and communicate with many different SSB or Ham stations or other vessels. There are many SSB "radio nets" that keep track of sailors in all parts of the world, and offer "phone & message forwarding" to family, friends & relatives.