The carpet is light in color, the base mouldings are white woodwork. Right?
Yes. The house is airtight. In the 80's and 90's, we spent so much time trying to make our houses energy efficient that now we have a building code which says a house has to be built airtight, then we have to install a mechanical system for exchanging inside air with fresh air from the outside.
This is not your carpet layers fault, nor is it the builder's. The fact is, your carpet does not get vacuumed at the edge by your regular vacuum cleaner, and this becomes noticeable when you have white woodwork and light colored carpet. There is nothing magical about berber; it shows every water stain, and god help you if you snag it.
The "ghosting" is nothing more than dust settling along the base of your walls, on the sides of your stairs, etc. It is not going to hurt your baby, who is more likely to eat dirt when playing outside in a year or two. Caulking the edges of the stairs is not reasonable, nor is it going to solve your "problem".
If air was "leaking" past the stairs, what's drawing it? Your stairs are not likely on the outside of the house, rather, they are probably completely within the exterior envelope. So where is the negative air pressure coming from? A situation like that is usually caused by a furnace that pumps more air into a room than can escape from the room via the cold air returns. This is highly unusual on a staircase.
I almost always agree with Jay J, but not this time. Just learn to vacuum a little better, bring the wand attachment from your canister vacuum out once in a while, and live with the latest in designer living. In a couple more years, we'll be back to darker colors of carpet and natural finished woodwork, and you won't see this anymore.