Travelers should take the same precautions that they would in a major city in the US. Pay attention to the advice of your tour leader and hotel reception and take common-sense precautions such as not going into unfamiliar areas alone, especially at night. Demonstrations, usually non-violent, are not uncommon in Ecuador. Travelers are advised to avoid these demonstrations as protestors occasionally burn tires, throw Molotov cocktails, or fire handguns into the air.
Petty theft is common in airports, crowded tourist sites, and busy markets so be aware of your valuables and avoid carrying large amounts of cash, jewelry or other expensive items. Check out the US State Department travel advisories for the latest information at: http://travel.state.gov/travel/cis_pa_tw/cis/cis_1106.html