detect non-domain computers in my network

you can join them by adding below line:
Add-Computer -ComputerName $computers -Domain "YourDomainName" -Restart

I would use radius to protect your network and only let domain computers connect

Scanning the network using ping  resp. Test-Connection requires that the default setting of the Windows Firewall is changed, as by default pings are not answered (at least for client OS). That's the first flaw.

The second flaw is that your WMI call checks the local machine, not remote. You need to include the computername parameter.

You should not only check if the machine is part of a domain, but of your domain.

Assuming ping works, your script would have to look like

$firstThree = '10.20.51'
$domain = 'mydomain.com'

1..254 | ForEach-Object {
    $address = $firstThree + '.' + $_
    If (Test-Connection -ComputerName $address -Count 1 -Quiet) {
        Write-Host  -f green "$address is up"  -NoNewLine
        $sysInfo = Get-WMIObject Win32_ComputerSystem -ComputerName $address
        if ( $sysinfo.partOfDomain -and $sysinfo.Domain -eq $domain )
        {
            Write-Host -f green ' and domain member'
        } else {
            Write-Host -f yellow ' but not domain-joined, trying to add now'
            Add-Computer -ComputerName $address -Domain $domain -Restart
        }
    } Else {
        Write-Host -f red "$address is not reachable" 
    }
}

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Carefully consider whether you want to use -Restart when adding the machine to the domain - this will force a reboot, which might be too intrusive.

You will probably need to add the following parameters to Add-Computer
-LocalCredential -UnjoinDomainCredential -Credential
If Add-Computer is anything like NETDOM, you won't be able to join another domain directly

Suggest change line 8 to add the -ErrorAction SilentlyContinue

 $sysInfo = Get-WMIObject Win32_ComputerSystem -ComputerName $address -ErrorAction SilentlyContinue

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