Every time I try to propagate a baby spider plant into a new pot from its parent, the babies die after a week. Their leaves turn yellow and fall. Their roots rot. Why is this happening?
I believe you are simply over watering the baby plant, or the soil you are using does not provide enough drainage. But there are a few ways we can work around that problem.
A Quick Background on Spider Plant Propagation
I can think of few things easier to propagate than a spider plant. Those little baby spiders hanging from the flower stalks (stolen) are each a complete plant with leaves, a crown, and aerial roots from which the root hairs will grow readily. A baby spider is almost a self-sustaining plant, so you don't need to submerge it in a lot of water like you might when rooting a stem cutting without roots of their own (although starting spider plants in water will often work, too).
The general idea is to simply clip that stolen from the main plant and place the areal roots in some moist potting soil mix. Avoid covering the entire crown or submerging the leaves of the plant below the soil level (this my be where your rotting problem is coming from). The new roots should grow quickly from the areal roots, so the daughter plant should do fine in the same growing conditions as the parent plant.
If you are still having difficulty with rotting, there's another method of propagating spider plants that may prove to be a bit more forgiving.
A different approach
Try leaving the baby spider attached to the parent plant and rest the aerial roots on a pot of well-drained, prepared garden soil mix. Push the aerial roots just a bit into the soil, but don't go too deep. You don't want to submerge the crown or leaves below the soil surface.
Again, keep the soil moist, but not too wet. Since the plantlet is still sustained by the parent plant, this gives you a bit of leeway with that "lifeline" if you don't get the watering right.
You're goal for the next few weeks is to simply keep the soil from completely drying out or becoming overly saturated. Don't overwater. In a few weeks, you should have a self-sustaining plant you can cut away from the parent.
Here is a quick video to see how it looks in actual practice: