Answer of Virtual Programming:
This seems to be an issue with Linux distribution, rather than with the game. Bioshock coordinates its game saving with Steam client, and it even is supposed to store the saved games on in the Steam cloud so you can share them between different machines you play them on, so issue with game saving is most likely something wrong between Steam client on Antegros and Antergos itself, not with the game. I recommend asking on Steam forums and googling for issues between Steam and Antergos to find the answer, and making sure that you have the latest Steam version installed, and that Steam client’s configuration doesn’t have unusual settings set, like disabling Steam cloud games.
Aprendiendo que no es poco