Avoid letting Bowser make more than three full rotations before throwing him; afterwards, he will spin too fast and become more difficult to aim.
In order to trigger the text after Bowser is defeated, Mario must be facing towards his tail and be within a short distance.
The "whoosh" sound that plays while spinning Bowser plays when he faces north, and is not indicative of whether he is pointing towards a bomb. Try to focus more on aiming visually.
It is highly recommended that runners who play using the original Nintendo 64 controller switch to a different controller for Bowser fights, as rapidly spinning the N64 controller's stick will wear it down. Recommended controllers include the Hori Pad Mini, a Retro-Bit Tribute 64, or a GameCube controller via a Raphnet adapter.
Bowser in the Dark World
Standard
To avoid an air throw, let the stick go neutral right before pressing B to throw.
Bowser has a ~10% chance to "dance" in place at the beginning of the fight; this allows you to grab his tail earlier and thus throw him quicker. Grabbing the tail near the tip places you closer to the bomb and can make this throw easier.
In this battle, Mario cannot grab Bowser until the platform stops teetering. Avoid groundpounding around Bowser's side of the platform during this period, otherwise Mario may clip through and fall into the lava underneath.