I honestly just use ziploc type bags of different sizes to store my pawns.
For example, I have one large bag labeled "Medium Monsters". Within that bag are smaller bags labelled "Monsters A-C" and "Monsters D-F", etc.
I have seen many beautiful and creative ways of organizing Pawns online, but the advantage of my system is that it is very easy to identify and pull out monsters on the fly. Cleanup is also very quick.
Many of the more beautiful storage systems I have seen online do not seem to be as quick and efficient as my bag system.
If you use a drawer-type system, you have to flip through the pawns and may have trouble seeing the pictures/names of Pawns. Cleanup/ reorganizing is also a bit slow. If you use a folder with plastic sheets, finding Pawns is quick, but cleanup is a bit slow, and adding new Pawns may mess up your organization system (e.g. if alphabetized). With my bag system, you just find the right bag (easy), dump it out (easy), and put the unneeded Pawns back in the bag (easy). Cleanup is equally easy.
I keep my bags in one of those plastic filing boxes with a handle on top.