See the Source link at the bottom for further info.
Owners of BioShock, BioShock 2, and/or Minerva’s Den on Steam will see the remastered versions of the respective games appear in their Steam library automatically. Those who purchased BioShock when it was new, however, either on disc or through Games for Windows, will have to take a few extra steps by providing your Steam profile information and proof of purchase of the original game to 2K Support. (More information on submitting a ticket is available here.) Once that’s done and approved, the remastered version of the game will be added to your Steam library.
2K confirmed that some proof of purchase is required, either a scan of your original receipt or a purchase confirmation email if you bought it digitally. That’s bound to be a stumbling block for some existing owners (including me), but a rep said that under these rather unusual circumstances, some kind of verification is necessary. The rep also clarified that both games will remain on Steam, so you can go back to play the original release if you like, but there will be only one Steam listing for the individual games, which will grant ownership of both editions.
Life is easier for owners of BioShock 2: Simply enter your Games for Windows Live key on Steam to add the game to your account, and the upgrade will be yours. (A detailed breakdown of the process can be had here.) BioShock Infinite isn’t being remastered for the PC because it already supports 1080p resolutions.