It wasn’t completely obvious so I’m making a post here for others who may be stuck.
Black Mesa requires the “Source SDK Base 2007” installed to run. If you are clicking on “Play” in Steam and it isn’t doing anything then that may be your issue.
To install the SDK, in Steam, in your “Library” tab click on the drop-down where it says “All Games”, “Installed Games”, etc. In that drop-down choose “Tools” and in that list your will see “Source SDK Base 2007.” Right-click on that and choose “Install Game.”
Once that is downloaded and installed you should now be able to launch Black Mesa.