Remove the cover hiding the ceiling fan's connections to the ceiling. You will either need to unclip or unscrew this cover from the ceiling fan. Make sure the mounting screws are tight and everything is intact.
Look at the fan blades to see if they are straight. If one blade is slightly askew, it can make the ceiling fan wobble excessively.
Measure the distances between fan blades. If the distance is not the same between each blade, adjust them to make sure the blades fit correctly.
Tighten the screws holding the fan blades to the motor. Loose or missing screws can cause shaking.
Check and tighten all screws on the ceiling fan itself. Normal wear and tear can cause screws to loosen or fall out. Even one small screw can unbalance the fan. A hardware store will likely have replacement screws.
Install a balance kit on the ceiling fan. Hardware stores sell kits that include a small piece of rubber you can place on a ceiling fan blade. Through trial and error, you can clip the piece of rubber onto a blade to help balance it when you turn on the fan.