Securing a MongoDB instance on Windows Azure requires attention to several key areas, including password management and endpoint security. A strong password for the Administrator user should be set, and the default username "Administrator" can be changed by logging onto the instance or using PowerShell. The installer creates three endpoints by default: RDP (3389), MongoDB (27017), and PowerShell remoting (5985), which can be secured by removing unused endpoints, choosing non-standard ports, and securing them to a specific location using firewall rules and Azure ACLs. Additionally, only allowing access from specified IP addresses or subnets can further enhance security.