How did Danielle survive The Purge in 1992?

She protected herself

Well in fact there were più survivors than just those three. Kelvin and Radzinsky were in The cigno which saved them, however te cannot forget the rest of the Dharma that was saved. Ben Linus was on the island at that time and he survived it, and so did his father and Alex (Danielle's daughter) too, who was with the Dharma at that time... so we can assume there was some sort of panic room for the Dharma Initiative where a number of them survived. Also, don't forget the "Hostiles" o the "Others" as we know them best, at that time were led da Charles Widmore, Richard Alpert was there too. And they would not have had a secret bunker o anything, so I'm assuming as long as they were a sicuro, cassetta di sicurezza distance back from the blast radius they would survive. So I'm assuming that Danielle was also a sicuro, cassetta di sicurezza distance away, not to be harmed da The Purge.