That line repeats a detail from the pilot, which shows Kara’s family sending her from Krypton to Earth. Her mother is a judge, showing how Kryptonian women could exercise authority.
In those scenes, Kara is about thirteen years old. Her cousin Kal-El is still a little baby. And their parents know that when the kids arrive on Earth they’ll have extraordinary powers—the yellow sun, you know.
But the women don’t tell Kara that she’ll have to be careful not to hurt the natives, or that she can be a hero. They say her job is “to look after your baby cousin, Kal-El.”
Kara replies, “I won’t fail Kal-El, or you.”
The grown-ups don’t tell Kara that she should do this “because you’re older, and he’s just a baby.” They don’t say she should do this “until he can look after himself.” They present this as her one and only mission in life.
The credits go on to explain that Kara’s spaceship was knocked off course, causing her not to arrive on Earth until her cousin has already grown up and become Superman. But she’s still in her teens. The series picks up a few years later when she’s in her early twenties. (Most comic-book versions of Supergirl are in their teens, but there is some precedent for a twentysomething.)
At first Kara says, “I didn’t have a mission anymore. But even though I had all the same powers he did, I decided the best thing I could do was fit in. After all, Earth didn’t need another hero.” Over the first couple of episodes, she decides to discard that attitude and be a superhero as good as her cousin.
Comics writers have gotten a lot of stories out of Supergirl’s wish to match her older, established cousin’s heroism. I have no problem with that being a big theme in this series. But here that theme is tied onto the notion that her job was to look after the family’s sole surviving male.