Children
Dylan and Patricia
To paraphrase Whitney Houston, we believe that children are our future.
And the cute little tykes in Independence Day most definitely symbolize that "tomorrow" that all the adults are fighting to save. Dylan (Jasmine's son) and Patricia (the President's daughter) put cute chubby faces on all the possibilities, lives, and futures that the aliens are threatening.
For example, an early dramatic moment in the film comes when Jasmine and Dylan are stuck in traffic in a tunnel when the explosions start. After seeing shots of several lone adults across the country who just kind of sit helpless in their cars as fire overtakes them, we notice that Jasmine's response is pretty different—she is not going to wait around and let some aliens kill her baby (or her dog, for that matter).
When she spots the fireball coming right for them, rather than being paralyzed, she leaps into action:
JASMINE: Oh, my God! Come on, Dylan! We gotta go!
Somehow, she notices an unmarked door that no one else has spotted (that's convenient) and busts it open in time to hide her, Dylan, and Boomer the dog before the wave of fire passes through.
Get it, Jasmine.
We're not sure, but it seems like Jasmine's protective impulses toward her kid help give her the extra strength, speed, and smarts to avoid the fire (where all these other poor folks are failing).
The stakes and emotions are just higher when you have a young kid around a dangerous situation—and that's probably why the film keeps Patricia and Dylan super visible to us, even when they're fading into the background of the action.