The rate of spoilage depends on bacterial contamination, temperature, and time. Keep your pot, lid, and utensils sanitary. Keeping the pot covered, and serving utensils sanitized reduces introduction of contaminates to the pot.
After the peanuts soften, I typically alternate simmering and soaking, until they reach the flavor and wetness I want.
You can store a covered, soaking pot of boiled peanuts in the refrigerator (below 40°F). Let them cool a bit on the stove, covered, then move them to the fridge. I reheat them later by bringing them to a rolling boil for about 15 minutes.
Boiled peanuts kept below 40°F are good for about 7 to 10 days. Frozen boiled peanuts are good for up to 1 year.