Have your net ready in your inventory. Shake a tree from BEHIND, this is crucial because it gives you the time you need to run away and get your net ready. When the hive drops out run NORTH a few squares, use Left or Right on the D-pad to pull out your net (it forces you to face forward, which is why I devised this strategy) and hold A to have it at the ready. When the gets close, right in front of you, release A and tada you just caught yourself a Bee!
Call me Kitty
Goal: 10 million bells earned from the stalk market
An alternate way to catch a bee is to stand to the north-west of a tree. After you see the beehive fall, you hit left or right on the D-pad, as you're pulling out your net keep presing A. This hasn't failed me yet.