I had to look into this LOL. It appears climbers actually go across the arch. From Wikipedia
[edit]BridgeClimb
Since 1998, BridgeClimb[38] has made it possible for tourists to climb the southern half of the bridge. Tours run throughout the day, from dawn to dusk, and are only cancelled for electrical storms or high wind. Night climbs are also available.
Groups of climbers are provided with protective clothing appropriate to the prevailing weather conditions and are given an orientation briefing before climbing. During the climb, attendees are secured to the bridge by a wire lifeline. Each climb begins on the eastern side of the bridge and ascends to the top. At the summit, the group crosses to the western side of the arch for the descent. Each climb is a three-and-a-half-hour experience.
In December 2006, BridgeClimb[38] launched an alternative to climbing the upper arches of the bridge. The Discovery Climb allows climbers to ascend the lower chord of the bridge and view its internal structure. From the apex of the lower chord, climbers ascend a staircase to a platform at the summit.
BridgeClimb celebrated its 10th anniversary on 1 October 2008, and is investigating the possibilities of other 'bridge climbs' around the world. Negotiations are currently underway for a similar operation on the Brooklyn Bridge in New York City.[39]
That sounds a bit iffy.
Diane