Sounding exactly the same, birth and berth are extremely different words and misusing them would lead to confusion…

birth: a beginning, especially relating to the beginning of life when a mammal leaves its Mother’s body
Parents see their baby’s birth as a turning point in their lives.

berth: a space for resting or sleeping, particularly relating to ships and trains. It can refer to a bed/bunk on a ship or train or a mooring place for a ship in a marina
The crew untied the ship from its berth on time for the departure.
Mary climbed into the top berth as the train sped past another village.

My tip for rememebring which spelling to use: a berth is a bed.