Syrup is sticky. It can be hard to open the jars easily if you store them in the fridge.
We keep it on the counter in a smaller pourable container. Ours is a little glass jar with a simple glass plug (the glass ring broke off nearly as soon as it entered our kitchen).
If you use the syrup every few days and disturb the surface by pouring some out with some regularity it doesn’t seem to mold even sitting on the counter un refrigerated.
Basically we keep the quart in the fridge and dispense as necessary into our table syrup container.