This was a major concern at the time, but wasn't publicized. It was dealt with in 1951, by attaching huge stiffening trusses under the deck.[1][2] If you've seen the underside of the Golden Gate Bridge, you've seen the trusses, which are not original. Prior to that stiffening, the bridge would twist in high winds, on one occasion enough that light poles hit the suspension cables.
[1] https://books.google.com/books?id=ug9YEcgBYs8C&pg=PA31&lpg=P... [2] http://goldengatebridge.org/research/majggbimprove.php