Drywall anchor won't hold
What you are using are the most basic drywall anchors available.  They're inexpensive and included in many products because of that, but they aren't good for much, and wouldn't be up to the task of holding something a toddler will be climbing on (you know they will...)

Look for products that spread out or lock into place when they enter the wall.  The name of the product might have the word "toggle" or "lock" or "twist".  There are several brands.  Here is one random example:



For the strongest hold (and also a larger hole that must be drilled in the wall) Toggle bolts are hard to beat.  I use them as a last resort because I've had great luck with plastic anchors and the hole these leave is large.  Of course for something as important as a stair gate, it might be worth it.