There is a difference between checking and gate checking. I am going to assume your child is over 2 yrs old since you are using a backless booster and hence would have a seat. If you want to check it just do so.
I personally would put it inside a large heavy duty clear plastic bag just to avoid any wear, tear, rain, spills, etc.
If you don't plan on using it on the plane I would just check it with the rest of the bags rather than hauling it down to the gate to check it plane side (aka gate check.)