Problem # 1: It usually makes a mess
The Fix: I suggest stripping the child to diaper/underwear, and place them somewhere that's not close to carpet.
Problem #2: The "masterpiece" is usually not something you want hanging around the house because..well...eegghhmm...it's done by a toddler :)
The Fix: This brilliant idea!! Make a Masterpiece out of your child's "masterpiece"!
picture courtesy of: squashblossombabies.com
What you need
Your Child's Art
Step One: Paint the entire canvas in your choice of color, let dry.
Step Two: Cut your child's artwork into shapes (circles, squares, etc.) and glue them lightly to your canvas in whatever design you prefer. Allow to dry.
Step Three: Modpodge the whole thing to secure the artwork and give it a a finished look.
Step Four: Dry and Done!!