has Liberty Island also been closed?
Re Hurricane Sandy - a Google search on "Liberty Island Hurricane Sandy" gives you this NPS page which says
"Both Liberty and Ellis Islands were largely submerged by the storm surge from Hurricane Sandy, which made landfall on the evening of October 28, 2012. Liberty Island reopened on July 4, 2013."https://www.nps.gov/stli/after-hurricane-sandy.htm
As regards 9/11.
There was the period immediately after the attack"Liberty Island, the statue's 12-acre home, was closed for 100 days after Sept. 11, 2001. Airport-type metal detectors were installed to screen visitors boarding the ferry from lower Manhattan, and the island was reopened in December 2001."
The Statue and Base remained closed for a further period of 3 years"After the 9/11 attacks, the federal government closed the Statue of Liberty National Monument, the symbol of a nation's dreams and one of the city's best-known and most-visited tourist attractions. Allowing people inside would be unsafe, the National Park Service said, because rescuers might not be able to get them out in an emergency."Three years later, the base, the pedestal and the observation deck were reopened after $6.7 million in improvements to fire and security systems."
During the Statue's refurbishment from 2011/12 Liberty Island certainly remained open - "the vast majority of visitors do not bother with the climb inside the statue. "They're going to get the exact same experience" while the statue is closed, he said, because Liberty Island will remain open while the work is going on."
( http://www.nytimes.com/2011/08/11/nyregion/statue-of-liberty-will-close-for-a-year-to -further-improve-safety.html?_r=0