I used The cross bow with grappling hooks, a long neck rifle, and shocking tranquilizer darts. I annoyed the Griffin by shooting it with a hook at close range. Very close, right under it close, once the Griffin takes off and your hooked to it, fire darts into it till it falls. The hook keeps you close to the Griffin. May have to repeatedly hook yourself back to the griffin, but as long as you do it just right you wont break from it till it's down. Once it's down I suggest blocking the Griffin with Dino Gates. You can also use Bio Toxin to keep it asleep. this is just the method I used to tame my first Griffin. (exploring other ways still)
Just build a 2x4 or 2x6 pen with a dino gate on one side, placing stone pillars on the outer edges all along it. Make it near the green obelisk! Then fly the griffin into it thru the door with a pteranodon, fly out thru other end thru the pillars, land your mount, run to the other side, close the door, boom. Trapped griffin. Then tame with roughly 150 tranq arrows. Use mutton or prime meat as they are easiest to come by.