"Secure" isn't a yes or no proposition. It's relative. Just like your house - you lock the door, but really - COULD somebody with a crowbar get in fairly easily through the window?
There are numerous articles written on how to use PHP. Given its open nature and its power, PHP is notoriously difficult to run truly securely.
Typically, it's slightly better to use POST rather than GET since a GET style form can be copied off the URL bar, saved as a bookmark, or emailed with little inconvenience.
Also, if it really must be secure, don't hesitate to set up SSL. Certs can be had for $50 and most ISPs will only charge about that much to set you up.
Then too, what about security of the server? Are there other users on the system that have the right permissions to read your scripts, logins, password files, etc? Are all relevant security patches applied routinely to your server?
Who can connect to your database? Are your passwords in plaintext, or are they encrypted with MD5 or something similar?
REMOTE_ADDR can be spoofed, though , circumstances depending, it can be difficult. Depends on how rigid you are about its use and your network topology.
If you want to secure your stuff, be careful not to pidgeonhole yourself in a certain way of thinking - the nature and breadth of attacks and compromises can be nothing short of stunning.
It's up to you to evaluate how much security is secure enough and plan accordingly. Remember that security is about restricting, and too much security renders a product useless.