In short, the Spider Grandmother has many worries just as Momaday's own grandmother has many worries.  Specifically, the Spider Grandmother worries about "raising the Sun's child" mostly because Spider Grandmother is part of the natural world while the Sun's child is partially divine. 

In order to understand this idea, it is important to understand a simple version of the Kiowa creation myth.  It begins with a boy born to a Kiowa woman and the Sun.  This makes the boy both human and divine.  Because the boy's mother cannot raise him, Spider Grandmother raises him.  It is during this time that her worries become paramount.  During her protection, the boy is split into two twins through a miracle.  One of the twins disappears into a lake.  By doing this, he becomes part of the natural world.  The other twin changes into the ten bundles and given to the Kiowa as a divine gift. 

As you can see, Spider Grandmother does insure the survival of the split twin sons, but one of them does disappear.  Spider Grandmother's fear, then, was unfounded.  It turns out that her connection with the natural world was important in that the one twin needed to return there.  In this way, Spider Grandmother becomes a representative of all Kiowa grandmothers:  a sign of tribal unity and tribal harmony.