Can you access the internet in Mojave, say via a browser? If no, then there's a connection issue to sort out. Suggest starting in System Preferences -> Networking, see what it says.
If yes, you might have the wrong Steam version for Mojave. Did you copy it from a later macOS version? Suggest going to steampowered.com and downloading the installer direct from Steam from within Mojave. It should install the right version for Mojave.
I just rebooted into Mojave, and launched Steam. Whenever it hasn't been run in awhile, it downloads an update. But on Mojave, it didn't do an immediate update. Once I got in, manually checking for updates said it was fully updated.
So maybe you need an older version of Steam for Mojave. My working Mojave Steam reports as:
Code:
Steam client application
Built: Mar 24, 2023, at 10:13:12
Steam API: v020
Steam package versions: 1679680416
https://www.steampowered.com
By comparison, my Sequoia Steam reports as API v023, main version 1769025840.