From what I can see, Bowser is weakest from below, as his Dair is horribly laggy, and Bowser Bomb is easily countered. On top of not falling extremely fast, this makes Bowser an excellent target for juggle traps. Capitalize on the Up ending of Dancing Blade and Uthrows to initiate.
Furthermore, Bowser's Jab and Dtilt are two hits, as well as hit Dair, UpB, and Dsmash having multihitboxes. Capitalize on reactive Dolphin Slash out of shield on these moves, BUT keep an eye on Bowser's percentage. He's the heaviest character in the game, and you don't want to Dolphin Slash hitting him nowhere, because that Fsmash hurts a lot, so make sure he's got a little hurt on him first, before using this.
Bowser is relatively easy to shieldstab with the down version of Dancing Blade, if his shield isn't full, so make sure you throw in Shield Breaker, to get in those hits, and get him off balance. Once Bowser is off the ground, he loses a lot of range, and his shield, so even a small hit can be critical if followed up well. Most characters are easy to shield stab with a tipper Dsmash (NOTE: YOU MUST TIP) if their shield is reduced enough, and tipper Dsmash hits up, where is exactly where Bowser doesn't want to be. If he's shielding, and his shield is kinda low enough to land this, try it.
Feel free to chase Bowser very far off the edge, since he really can't compete with Marth's sword and aerial maneuverability. Just remember to come back soon enough to initiate edge-guarding, and reset the situation. If you don't have the spacing to hit Bowser out of his UpB before he reaches the edge, then hog the edge, and then ledgehop Uair him while he's lagging from his UpB. This will start the juggle trap. Of course, optimally, between Fair, Nair, Dtilt, and Counter, you want to just keep him off the map once you get him there.
Also, I'll just throw out the reminder that Bowser has a hilarious Footstool animation, and is horribly gimped by Footstool since his UpB is not good at all for vertical recovery. If the chance to use this does present itself, go for it.