You simply have to find, buy and equip the Baby Carrier and play through any level wearing it. The idea behind this achievement is that the Baby Carrier adds nothing to your stats and the backpacks are very useful items.
Obtaining items is random so there's no quick or easy way to do this. You find items in the game world that unlock items in the shop (they're color-coded crates with icons on them letting you know what kind of item you just found). So just keep playing until you find and can afford this item:
Baby Carrier (Level 4, 1,170 nuts)
As you can see, this is basically a freebie achievement if you disregard the fact that finding the Baby Carrier is entirely based on luck. I didn't find it until I was level 28, but once I did, I obviously had far more money than needed and beating a level with it is not as much of a drawback as using the astronaut's gear.