Keep in mind that cooking destroys those good-for-you probiotic bacteria. Try chopping the fermented veg and adding them to grain-based salads such as tabouleh or chop them small and use them in place of capers. If you do decide to add them to a cooked dish such as soup, add them at the last minute after you've turned off the stove.
It is crucial that the vegetables remain below the brine so that they are not exposed to oxygen.
You can cover the jars with something breathable, like a paper coffee filter, a tea towel, or several layers of cheesecloth. Use a rubber band to hold the cover in place.
Alternatively, you can use a tight-fitting lid, but you will need to open the jar every day to let some of the gasses escape.
Another option is to purchase special fermentation lids that have a one-way valve that allows carbon dioxide to escape while preventing oxygen from entering the jar.