The Atlantic hurricane season officially starts on June 1st and ends on November 30th. The peak months for hurricanes are August to October, so the chances of being caught in a hurricane at the beginning or end of the season are lower. However, major hurricanes only occur every ten years or so. Beaches in the southeast of the islands, such as Punta Cana, are popular tourist destinations.
If you're looking for a peaceful visit, avoid going during spring break in the US (which usually lasts a week and falls between March and mid-April). The best time to visit Punta Cana is between November and February, when the weather is pleasant and the area is busiest. Punta Cana has a similar climate to Puerto Plata and an average temperature of 28°C in September.