Mars Attacks! the film was based on Mars Attacks, a set of trading cards (aka bubblegum cards) published by Topps in 1962.
The martians in the film are depicted with helmets because that's how they're depicted on the trading cards. Any other explanation is going to be an after-the-fact rationalization.
You can see images of the entire set of cards here. There's a paragraph on the back of each one explaining what each one was. None of the text directly addresses why the martians wear helmets on earth.
Cards 48 and later depict Martians on Mars living in domed cities, which could indicate that the Martian atmosphere is no longer ideal for their needs. But Martians are also shown outside the cities without helmets, so they can apparently operate in the martian atmosphere to some extent.
Humans are shown as needing breathing gear on Mars (with flak helmets on the inside, naturally), except for card 52 which is inside a Martian city.