The scaffolding fell onto 15 cars shortly before 11 a.m. in a construction area near vehicle inspection booths.
Eleven people were injured in the accident. A construction worker was in serious condition, said Maurice Luque, spokesman for the San Diego Fire-Rescue Department. The others were taken to hospitals with minor injuries.
All 24 of the border crossing's lanes entering the U.S. are now closed.
It is unknown when the San Ysidro crossing will reopen.
The 30-year-old border crossing facility is undergoing a $577 million overhaul by Hensel Phelps Construction Co., based in Greeley, Colo. Jeff Wenaas, the construction firm's chief executive, did not immediately respond to a message seeking comment.
The Associated Press contributed to this report.