Refrigeration for croutons is a bit of a mixed bag. Generally, croutons can stay crispy and fresh in an airtight container at room temperature for a few weeks. Keeping them in the fridge might prolong their shelf life slightly and keep them crispy, but if they're already crunchy and well-stored, it's not strictly necessary.
If you live in a humid climate or if you want to be extra cautious about freshness, popping them in the fridge is a good idea. Just make sure they're in an airtight container to avoid them absorbing any moisture or odors from the fridge. For most people, though, storing them in a cool, dry place should be enough to keep them crispy and delicious!