Azores along with continental Portugal has a very low crime rate, and you should not run into any problem whilst traveling throughout the area.

As with any locale you are unfamiliar with, it pays to be alert while traveling within large crowds, especially during the high season. Make use of your hotel's safe to protect items such as any jewelry you might have brought with you (which, in reality, is best left at home), extra monies, credit cards, and passports. Keep your eyes and a hand on bags at all times, and do not leave your items unattended. When visiting the beach, take special care of any valuables that you bring with you. Stuffing a wallet in the back of a shoe is not a great way to avoid having items stolen. If you choose to head into the water, have someone remain at the beach to watch over your belongings.

In addition, when traveling abroad, it is often easiest to travel with traveller checques, which are easier to replace than cash or credit cards if stolen. Keep a copy of the front page from your passport, any tickets, traveller checque serial numbers, and any other important information in a location separate from the originals. You may also consider leaving a copy of these pages with someone back at home, whom you can contact if necessary if you get into trouble abroad. These copies will come in handy if you are a victim of theft.  

The latest travel information for the Azores can be found on the US Department of State Consular Information sheet for Portugal.