I went last Wednesday night/into Thursday morning (rush for Thursday 5/30 performance). I arrived at 12:30am and was the only one in line until 4am, when two more people arrived. There were tourists walking back and forth most of the early hours of the morning, and some people smoking weed next to their car parked across the street for a while. That was about it. I felt very safe.
ALSO Reeve Carney was coming back from the bars at 4:15am with a couple of friends, and stopped by to thank the 3 of us for waiting in line to see the show! So that was surreal.
Afterwards, 2 more people came to wait at 5am. Another at 5:15. 2 more at 6am (so we were up to 8 in line). More came in a steady flow after that, and the line was crazy long around the block by 10am, but only those of us there by 5am got rush-priced tickets. The ones next in line got the $99 seats and some waited for SRO, which they didn't release until the evening. One of the women who was there to clean working for the theater told me they were consistently giving out 7-12 rush tickets each performance, but not more than that.
It was a long time to wait, but SO WORTH IT to see this show before the Tony's! It's incredible. Best of luck!