How can I secure windows which have been bypassed in my security system because they're open?
If the existing sensor is a surface mounted magnet / reed type, it's very easy to add a magnet.  You want the sensor about halfway up the window, and two magnets on the window:  one down at the bottom corner, and one halfway up, same spot where the sensor is.  This way the magnet will operate the sensor in two positions.  

I say half way up just because that's usually a spot where it will be hard to crawl through the window but you still get some air.  That may or may not work out for you depending on the position of the window etc.  You could set up the sensor and the magnets so you can open the window just a few inches or open wide, and still arm the system.  You get the idea.  

Alarmed screens are more expensive but let you do whatever you want with the window - it no longer matters whether the windows are open or closed.  You can send the screen you have out to a shop that adds a wire trace to the screen.  The sensor has to be changed as well, which can be some work with a hardwired system, depending.