The iPhone SMS alerts don't offer a lot of configuration. You're always going to get an alert when you get an SMS message, even if phone is logged off. But the current SMS alert also includes information about the sender and a preview of the message. I don't know about you but I which I could turn this off as I don't necessarily want anyone who happens to be looking at my phone seeing this sort of information.

While Apple doesn't let you turn the messages off, you can at least configure your phone to not show the sender and message preview when you are not logged into your phone. You can set this by going to "Settings" => "General". If you turn the passcode on, you can then set an option for the "Show SMS Preview" and you can turn this off. When it's "off", you still get an alert, but it doesn't show any details. And it's only off when you phone is locked. But it's better than nothing.