I walk through airport security with a mechanical watch on almost every other day. If it has any impact, I haven't noticed it. Rarely, I've had to remove a watch -- when they have the scanners turned to super-high sensitivity -- and put it through the x-ray machine. I haven't noticed any issue the either.
I know people encounter electromagnetic fields all day, everyday, but the scanners at the airport emit radiation, not electricity, so whatever electromagnetic field those machines create should not differ whether they are scanning or not, so long as they are turned on. If walking on the non-passenger side of the machine doesn't affect the watch, walking through and being scanned won't either, at least as far as electromagnetism is concerned.
Think of it like this...light, electricity and magnetism are just the evidence for one thing: electromagnetic radiation. Would you be concerned about shining a flashlight at your watch? The x-rays and such that scan travelers at airport security are essentially just a big flashlight that emits light rays that aren't visible to our eyes. That's the "quick and dirty." To get the full story, read this:
http://www.astronomynotes.com/light/s3.htm .
All the best.