I have created a file share in Microsoft Azure. I have looked but not been able to find if there is any way to create permissions on the file share created. Or is it a open file share where anyone who has access can view everything?
You could use shared access signatures (SAS) to give permissions to your Azure File Share.
A shared access signature (SAS) provides you with a way to grant limited access to objects in your storage account to other clients, without exposing your account key. More information about SAS please refer to this link.
Azure File storage service is not designed to provide external users access to files. It's primary purpose is to act as a file share for cloud services and virtual machines running in Azure.
You should use Shared Access Signatures to share files with external users.
Azure just enabled a preview of Azure File Share Sync. You can sync shares on premise to Azure File Shares and permissions are transferred. Then you can setup a new server in Azure to make available to users with permissions intact. I put this blog post together on it.