I'm not the person you asked, but IME it should be closed, airtight, in fact, would probably be the best way. Otherwise you risk spores / molds etc. getting in and spoiling it, not to mention insects and other tiny creatures, if they are around in the environment, even inside a house, and some often are, unless your house is a sterile, sealed environment.

I agree with the other responder--it's better to have the jar covered. However, fermentation produces CO2, so you can't have it sealed. Fermentation crocks have a water ring for the lid to sit in so excess gas can bubble out, but you can also just put a jar lid on but not screwed tight.