The phrase "have a fling at" implies taking a chance or trying something unfamiliar. Its antonyms would be words that convey reluctance, hesitation, or aversion to taking risks. Some antonyms include "play it safe," "hold back," "restrain oneself," "eschew," "abstain," or "avoid." These antonyms emphasize the importance of caution and prudence in decision-making, rather than impulsivity and recklessness. Taking calculated risks can be beneficial in some situations, but it's crucial to balance them with careful consideration of the potential consequences. Being mindful of the antonyms can help individuals make more informed choices that align with their values and goals.