You must gain their respect before trying to tame one otherwise you'll end up using your soothing balms and other resources in vain as they only start taming once you have their respect. To do this you must atleast be at lvl 85 or above. You can also use a name change ticket to make your name same as the Griffin you wish to tame as this will earn their respect. Or you could buy the gold crown available in the shop.

More Royal Griffin Taming & KO Tips