The best official way is to play the compilations. Capcom Classics Collection Vol. 2 for the PlayStation 2 (which is backward compatible with original PS1—no, wait, PS2 is not backward with PS1? Actually, the PS2 can play PS1 games, but this is a PS2 disc). More relevant: For PS3, the Capcom Arcade Cabinet downloadable title included Cadillacs and Dinosaurs in some regions (though it was delisted). This is not a PS1 ROM, but it is official Sony hardware gameplay.
Because other Capcom arcade hits like Street Fighter II and Final Fight made the jump to home consoles, fans naturally expected a PlayStation 1 (PS1) or Saturn port. However, . Due to complex licensing issues involving the Cadillac brand and the comic's creator, Capcom never officially ported the arcade original to any console. Cadillacs And Dinosaurs (Arcade/Sega CD) : r/SegaCD cadillacs and dinosaurs ps1 rom
To directly answer the core question that drives thousands of emulation searches: Capcom never officially ported or released Cadillacs and Dinosaurs for the Sony PlayStation (PS1). The best official way is to play the compilations