I seem to have the opposite problem from most guitarists. I don't have much problem with my fretting hand (finding the right chord shape and/or fretting the correct note), but I am terrible at picking the right string with my strumming hand.

When I watch other guitarists, they're usually looking at their fretting hand to make sure they get it right - not me, I'm looking down at my picking/strumming hand. As long as I'm looking, I do okay. But as soon as I look somewhere else, holy cow.

Any advice or exercises? I've tried just playing some arpeggio over and over, but I seem to be no better after the 500th play-through than the first.
I guess everyone is different. I find looking at my picking hand makes me play worse. Like I'm trying to figure something out. When I should probably just go slower and build up speed. I'm still a relative noobie. Even a decent speed is a lot to visually process. The hand just needs to be semi on auto pilot. I can kind of use my brain to anticipate the areas I usually screw up, but beyond that it's mostly feel and practice, for me.
You need to be doing finger exercises with strum exercises, sounds like you don't practice technique enough.  Strumming and hitting the correct strings is directly related with the time spent teaching your brain and building muscle memory.  

Practice more, do actual practice stuff, not just learning new songs, chords, noodling.  Get a teacher if you don't know what I'm talking about.
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