The creature with Magic Resistance would have advantage on all subsequent saves against spells and other magical effects.
Taking from this answer to a similar question, if a creature has to make a save due to a spell, or magical effect, it would have advantage on that save.
Magic Resistance. The {creature} has advantage on saving throws
against spells and other magical effects. (MM, standard phrasing for magic resistance in the stat blocks)
From the specific example you gave with Bestow Curse, this would also apply to all subsequent saves if it fails. The curse bestowed on the creature by Bestow Curse is magical, as per this answer, because it is a spell. Since the curse is magical, they are continuously making a save against the magical effect of the curse to resist its effects. This means they would have advantage on the save against the curse's effects.
None of the other suggested Bestow Curse examples require additional saving throws, and would probably be better to use against a creature with Magical Resistance.