They’re looking for cracks and damage from the earthquake awhile back.
So, my question is, how do they get the ropes up there in the first place so that they can rappel down?
EarlyMan
Picture accompanying this article shows a small access window within a few feet of the top: lean your torso out, lasso the peak with your climbing rope, and you’re good to go.
It looks like they wrapped ropes around the monument at the upper window and dropped ropes down from there. Lassoing the peak is probably not the way you want to go. 
The natural sassy answer is “Very carefully!”
In reality, the job is being done by the “difficult access team” from engineering firm Wiss, Janey, Elstner Associates Inc. of Northbrook, Ill. I understand these fearless people are also members of the Society of Professional Rope Access Technicians.
I can only assume that there are two access hatches at the tip, so one person can tie off to something inside, lean out and pass lines around to someone else at the hatch on the other side.
Maybe YOU’RE good to go, but not me, pardner. <nauseated smiley>
I was actually hoping it had a flip top and opened like a giant PEZ dispenser. 
This video shows how they went out a small hatch near the top and rigged a harness from which to hang lines down each side. Subsequent climbers can go out through the larger windows lower down on the structure.
This is the first time anyone has ever rappelled down the outside of the monument.
I seem to recall that when they finished building it in the 1880s they topped it off with an aluminum pyramid (aluminum being considered a precious metal in those days). I wonder if that pyramid is still up there.
If the peak can take the weight of the rappler without failure, then the crack can’t be too serious and we can just fill it with silly puddy or bubble gum. 
I bet that French guy, Alain Robert they call him spider man, could climb it free hand. Compared to his other climbs the Washington Monument would be simple. He even climbed the Sears Tower.
It is.
Motto: Once you go SPRAT, you never go SPLAT!