湿らす (or its intransitive counterpart 湿る) is a matter-of-fact verb that means "to make/become (mildly) wet".
濡らす (or its intransitive counterpart 濡れる) is "to make/become (hardly/drippingly) wet".
潤す (or its intransitive counterpart 潤う) is used with a narrower range of objects that are considered "unhealthy" when dry, e.g., skin, lip, throat, meat, earth. In other words, 潤す means supplying water to something and putting it into a healthy moist status. The kanji 潤 has a clearly positive connotations (see: 潤い). 潤す even means "to (financially) enrich", "to (psychologically) comfort", etc., too.