In the new ARM world and with the latest PowerShell, you'll need to make some adjustments to @Seth's answer.
Specifically, the way you obtain the publishing creds is different, which is the first 3 lines. The rest I shamelessly copied from @Seth to complete the snippet.
Make sure to replace YourResourceGroup/YourWebApp as appropriate:
$creds = Invoke-AzureRmResourceAction -ResourceGroupName YourResourceGroup -ResourceType Microsoft.Web/sites/config -ResourceName YourWebApp/publishingcredentials -Action list -ApiVersion 2015-08-01 -Force
$username = $creds.Properties.PublishingUserName
$password = $creds.Properties.PublishingPassword
$base64AuthInfo = [Convert]::ToBase64String([Text.Encoding]::ASCII.GetBytes(("{0}:{1}" -f $username,$password)))
$apiBaseUrl = "https://$($website.Name).scm.azurewebsites.net/api"
$kuduVersion = Invoke-RestMethod -Uri "$apiBaseUrl/environment" -Headers @{Authorization=("Basic {0}" -f $base64AuthInfo)} -Method GET