I’ve already finished Bioshock on my current computer, but I felt like replaying it, so I downloaded it again. However, when I try to create a new save, the game crashes. After looking into the issue, I discovered that the crash always happens when the game autosaves—I should point out that when you try to start a new save, before the introduction scene on the boat during the storm, there’s an automatic save. I also found a crash log that might help understand why this is happening. Lastly, I’d like to mention that my hardware setup is the same as when I first completed the game, so I can rule out hardware issues (other games run fine, except for Deus Ex: Human Revolution, which shows the same problem).

Gog Galaxy client is fucking cursed. You can not save at all if you launched the game through Gog. Click on BioshockHD.exe that is in 'GOG Galaxy\Games\BioShock Remastered\Build\Final'. Crashing when saving will be gone.

Gog has no QA for simple shit like this.

Small update:
It seems the overlay feature, a fucking decade old problem, is still plaguing games. Everything seems to be working fine if the overlay feature for BioShock is disabled.

Manage installation > Configure...
Features: uncheck overlay

Another update:
Saving will sometimes crash. Make sure not to overwrite a save so you would only loose progress and not a save file.