Weekdays are the best time to visit if you prefer having a comparatively laidback experience with the dolphins. On weekends, you'll find a larger crowd at most Singapore attractions since offices and schools are closed. That being said, even on weekdays, plan your visit for earlier in the day since it will be more crowded as the day progresses.
Peak season lasts between December to January and again from June to August. During this time, the weather is mostly sunny and humid, which are the best conditions for outdoor activities like Dolphin Island. But be prepared for larger crowds at most tourist spots, along with longer wait times than during off season.