Is it a bad idea to keep a cut open onion in the fridge?

That would depend on how they have been stored and for how long. A very famous outbreak of botulism was the result of eating grilled onions that were stored covered in oil and placed beside the grill. The spores of Clostridium botulinum germinated and grew, then formed the toxin under the anaerobic conditions.

As a general guideline, cooked food should not be left unrefrigerated for more than 2 hours - or only one hour if the ambient temperature is quite warm.