Antonyms for "having a fling at" include taking things seriously, showing restraint, being cautious, and displaying maturity. When one is having a fling at something, they are typically engaging in reckless behavior or indulging in a momentary pleasure without much thought or consideration for the consequences. Antonyms, therefore, are words that signify the opposite of such behavior. Individuals who take things seriously are careful and thoughtful, while those who show restraint exercise self-control and avoid impulsive actions. Being cautious means being careful, while displaying maturity means acting in a responsible and wise manner. All of these antonyms promote responsible and measured behavior, rather than impulsive indulgence.