Would it be likely to be seen as odd or suspicious for someone who is not a pilot to stand outside the fence around a GA airport in the US watching the aircraft?
I mainly fly in the UK and Europe, however I trained in the US and have flown there a few times since. I have probably 50 hours in total in the US.
I can tell you that almost without fail, every airfield I have ever flown from anywhere in the world has always had some level of plane spotting activity. It is a hugely popular pastime with both photographers and people who just like watching.
I don't think there's anything particularly suspicious about it, so long as you're not trespassing, not touching anything you shouldn't be, and not causing any interferrence.
It depends on how you go about it.
Don't wear all black or camo with a face mask. Don't sneak around in the grass, or run from building to building seeking cover.
Plane spotting is a somewhat common hobby -- you might check out some plane spotting forums for tips on not getting hassled. You might think having a camera with a lens as long as your arm would be suspicious... yeah, it is still suspicious.
Many GA airports have a restaurant or a designated viewing area. My local small airport (KSBP) has both of these. It also has a picnic table along a back road near mid field at a particularly good vantage point. At the other end of the field, there is a wide spot in the road directly under the ILS approach path that people frequently use.
You should be able to find some contact information for an airport -- either a local airport authority, an FBO, or some other on airport business. Give them a call, they can probably tell you the best vantage point.
Of course, the best place to watch planes on approach at my airport is from one of the wineries under the approach path. Tolosa has a lovely garden with some seating areas. You can take your tasting outside and watch airplanes land from a couch. I realize that is not a solution that works for everybody, but it is good if you can get it.
Being from Arizona, the only major airports I know of are Gateway and Sky Harbor. Both of which have plenty of plane watchers. As a previous answer has stated, don't touch anything, don't be sneaking around, and don't suddenly dart away as soon as you are seen, as people may think you're planning to hijack a plane, escape security, or commit a crime. Try not to wear all back or camouflage clothing, especially if you do happen to be sneaking around, darting off, and touching stuff, as that will make you even more suspicious. Other than that, most people aren't suspicious of plane watchers.
EDIT: As for GA airports: the only one I know of is in Iowa and I barely remember going there (I was about 2 or 3 at the time) but according to my father, the fences surrounding the airport were lined with plane watchers so I'd think from that information it's totally fine.