Is there a way to keep the screen steady in Smash Brothers?

If you look back at the N64 version, the screen hardly ever shakes/rumbles/tremors. I believe it only shakes during bombs and some select heavy attacks.

In the Melee and Brawl versions, however, it is almost non-stop. I have always found it annoying and that it made it hard to play long sessions and hard in general to track players. This is especially true when two people are beating each other up on the other side of the screen, the camera is zoomed out, and I am trying to fight someone else. When I do a 1v1 the screen shaking basically doesn't bother me.

You can see the difference on this smash bros video at 9:53 and then 10:26.

Is there an option or a cheat to turn off screen shaking?

For Melee, the 20XX Melee training hack pack offers the option to disable screen shake.

I don't know about the other games in the franchise.

To be honest, I've never heard a complaint about the screen shaking before - most people I know (mostly non-gamer relatives) don't even notice it until it's pointed out. You might just be one of the unlucky ones who can't handle pseudo-unpredictable shaking as well as other people can.

There is no in-game option to change screen shaking in any game. There may or may not be hacks to do so, but I'm not aware of them.