steam_api.dll
This file is not a system component: supplied with the game by steam. Restore it from the source that installed it rather than downloading a replacement.
Do not download this DLL on its own
Searching for a missing DLL by name leads to sites that hand you that single file. It is one of the oldest and most effective malware delivery methods there is, and a system DLL from an unknown source is indistinguishable from a deliberately modified one. There is never a legitimate reason to obtain a runtime component that way.
What it is
The Steam API library. It is not a system component: it ships inside each game folder and is installed by Steam along with the game.
Where you see it: A game that fails to start after an interrupted update or a file that got quarantined.
If it is missing from a game you own, use Steam's Verify integrity of game files and it will be restored. It is worth knowing that this error is also extremely common with unauthorised copies, which is why so many sites offer the file directly. Downloading a replacement steam_api.dll from a random site is one of the most reliable ways to infect a machine.
Why there is nothing to download here
There is no verified download we can offer for this one, because installing a file is not the fix. The steps above are what actually resolves it. That distinction matters: the sites ranking for this file name are selling a download that would not have helped even if it were safe.