Constanţa is a pretty safe city even by European standards, but as a precaution avoid taking taxis near the train station and insist that they turn the meter on. If you visit the historical peninsular area at night, try to do this in a group. Unlike some other cities in Romania, people from Constanţa are well used to foreigners and are generally helpful. Walking at night is okay even alone, though you should avoid walking with your camera around your neck and be mindful of pick pockets when using public transportation. Your religion is not of concern since a lot of locals come from very different ethnic and religious backgrounds, but refrain from commenting on religious matters. Unlike Western Europe or the Middle East, the local Muslim population is pretty secular and don't have a problem with western culture or habits.
Stay away from dodgy small casinos and avoid exchange offices especially those located in Mamaia. They have hidden commission written with small letters or separate exchange rates for travel checks and cash. Use only trusted exchange offices, like Balkan Exchange. If you are unsure go to a bank, Your best bet is to exchange money at banks, which are located throughout the city. Although banks don't have good exchange rates, they are still better than the rip-offs).
As with Bucharest and other big cities in Romania your biggest safety concern could come from feral dogs. If confronted by a group of barking dogs don't loose your cool and back away since they are usually territorial and won't chase you. If they get too close, pretend you are picking a rock from the ground. Whatever you do, don't run.
Women will not be permitted access in churches or mosques if they wear short skirts.
