I understand your concerns. I've been across the bridge several times, but not with my wife. She just won't go, either afoot or in a vehicle. The only serious bridge she's ever crossed with me is the one at the Glen Canyon Dam in Arizona, and I had her down on the bridge before she knew it was coming. I got a serious smack for that!!!
The Rio Grande Gorge Bridge is spectacular and fascinating for me. Yes, it moves, especially when a vehicle or more is crossing it. Yeah, the wind blows sometimes pretty seriously across it and the canyon as well. And yeah, it's a looooong way down. But that bridge ain't goin' anywhere, except of course across the canyon. I know it's easy for me to say, since I don't have a problem with it, but ya gotta tell yourself that it is safe, because it is! Maybe you'd be better off not to ever look down from the bridge, either riding across or on foot. If so, then don't look to the right or left either, up and down the canyon. Just ride right on across and maybe look after you get across, unless of course you plan to come back the same road to get home!
It is a beautiful place ... the road leading up to the gorge from the east will cause you to wonder how such a thing could be there. I can only imagine what the folks riding a covered wagon must have thought when they pulled up to the edge after crossing that wide open flat area to the east of the gorge. They had to go a long, long way to find any place where they could continue their journey westward!! I think you'll be glad to have seen and experienced this place.