I have a study that is 9 x 11 with 12' ceilings. What is the best way for me to shorten the height of the room without taking away from the square footage? Should I install chair rail?

Most moulding is used to make rooms look taller. Why are you trying to shorten the height of the room?

You Can Start By Using Chair Rail At 36" But The Best Way To Get The Ceiling To Not Look So High Is To Use Crown Molding. At 12 Ft Ceiling You Need At Least A 5 1/4 Crown And Maybe Some Basecap 2" Below That. We Trim Tall Forures Like This All The Time. Good Luck , Nichols Trim