If Runtime Resources show as offline within Control Room which is running directly on the Application Server (whether connected directly to a database or connected to a Blue Prism Server type connection), then you need to check the communications between the Runtime Resource and the Application Server.
Ensure that there are no firewall settings (both hardware and software) blocking the ports in use, and also the applications themselves (automate.exe and for the app server - blueprismserverservice.exe or BPserver.exe depending on whether you are running your BP server via the server service or not). Also, check that there is no Network Address Translation in use between the machines (v4.2 and v5, and v6 if using .NET Remoting type of configuration).