Can anyone give me some tips on how (in Adobe Premiere or After Effects) to get that look of a videocam or security camera.

The look I have in mind is similar to near the start of "28 Days Later" or the videocam in "I Spy", where the video looks as if it has light and dark horizontal bands.

Not sure if I have explained the look clearly... but basically a degraded vertical resolution to the image.

http://www.adobe.com/products/aftereffects/imgeffects.html

http://www.adobe.com/products/aftereffects/imgeffects.html
wow those presets in 7 look great

Those light and dark bands are meant to emulate a monitor that's out of phase. This can be emulated by (under controlled circumstances) recording a television screen. Use any Clear Scan funtionality you might have in-cam to adjust the scroll of the lines.

Or use a plug-in, of course.

Sony Vegas has a nice filter for this called "TV Look." Give that a try if you use vegas.