DST Fails to Start. SteamAPI_Init() Failed

Generally the "Fatal Error: Steam must be running to play this game (steamapi_init() failed)" error message stems from within the Steam client. Your best bet for getting technical support for Steam client/Steam API issues is to contact Steam Support directly.

In the meantime, here are some troubleshooting steps that you can try that may be of help:

1) Fully exit your Steam client, then fully restart your computer. Once restarted, run Steam once more and see if the games work now.

2) Ensure that you are running your Steam Client as well as Don't Starve Together as an administrator. Setting up administrator access for PC: https://support.microsoft.com/en-us/help/14028/windows-7-how-log-on-as-an-administrator (Windows 7) OR https://www.cnet.com/how-to/how-to-check-if-your-windows-10-account-is-administrator/ (Windows 10)

3) Ensure that your antivirus program or firewall settings are NOT blocking your computer's communication with the Steam servers. Try temporarily disabling them to see if it fixes the issue.

If none of these help, your best, next step is to contact Steam Support (https://help.steampowered.com/) and inform them of your issue, provide any screenshots of the error message, and Steam Support will be able to help get that sorted for you.