The Atlantic hurricane season officially begins on 1 June and ends on 30 November. August through October are statistically the peak months for hurricanes, so the chances of being caught in a hurricane at either end of the season are reduced, but still a risk. Note that hurricane season runs from June to November, and the stormiest months tend to be August and September. However, tourists need not worry too much, as major hurricanes only occur every ten years or so.
Beaches in the southeast of the islands, such as Punta Cana, are popular destinations, but if you're looking for a little peace and quiet during your visit, think twice about going when it's spring break in the US (which usually lasts a week and falls sometime between March and mid-April). So when Punta Cana has its best weather, between November and February, it's not surprising that it's also the busiest time. Punta Cana has a similar climate to Puerto Plata and an average temperature of 28°C in September.