The cautious Fire Breath should really only be considered in the case of sword-wielders and maybe projectile users due to the ease they can out-prioritize you. Yoshi counts as both for purposes of priority and annoyance at the ledge.
After that, it depends on how dominating the character is off-stage. Bowser is very middle of the road when it comes to his off-stage tools. This is better than it sounds. In most cases, you should be looking to wall your opponent with the threat of a FAir, UAir, or BAir. Which tool you get to use largely depends on your opponent. FAir is more effective for sudden aggression (big hit box, disjointed). UAir is great for catching characters that double jump or attempt to send out a projectile at a trajectory you can avoid by traveling under. BAir is extremely effective if the opponent either doesn't have a response or makes an execution error (ie. mistiming a jump or air dodge). BAir is also easy to aim if you grab the ledge first for positioning.
In the case where you need to just keep calm and maintain stage control, your go-to-options should vary between down-angled FTilt, DTilt, Bowser Bomb, ledge trump, pivot grab, or a SH aerial in case they jump from the ledge. Be aware that Bombing to the ledge is an all-in gamble, as it will reverse your positions if you fail to hit. Of course, if they (can) challenge you, try experimenting with Shield -> USmash or Fortress. In the case of weird things like Ike's Aether, you could even experiment with just double jumping into a ledge canceled Bomb. This sounds dumb, but most things that work with Bowser are kinda dumb.
For an idea of which characters dominate off-stage, ask yourself if the opponent has a significant mobility option (ZSS, Sheik), super armor (Charizard, Yoshi), is named Jigglypuff, or possesses a way to outspace you with high priority rising hit-boxes (Ness, MK, Yoshi).