I spend a week by myself in Santo Domingo, work related. I stayed at the Hotel Jaragua downtown, which was a very nice luxury hotel. Arriving at the airport was scary - very crowded, very small building, and people trying to get money from you. Definitly watch your bags and purse there. Thank god I had a driver pick me up and take me to the hotel. Saw a lot of poverty driving from the airport to Santo Domingo.
At first I was reluctant to leave the hotel area to walk around by myself, since as soon as I stepped a foot outside, people - mostly young man - would come up to me to sell me stuff or talk to me. Of course having blond hair - I looked very touristy. But then I realized that the people are very friendly, and they will leave you alone after a firm "no." I walked around and explored the old downtown by myself. I would not go alone in the evening though.
Having said that, as long as you are in a group, and know where you are going, it's a great place and all the people I met were very friendly.