Where are you from? DH and I both are frequently in Baltimore, MD, and we felt fine in Nassau. I wonder if some of the people who are especially nervous to be out and about in Nassau don't spend very much time in big cities.
Yes, there are people trying to sell you things. Just say "no thanks" and keep on walking, or just ignore them and keep on walking! We decided to take a taxi over to Atlantis and get passes to see the aquariums (interesting, but we'll probably skip it next time), and then we went back to the shopping area, which is right at the cruise ship dock, and explored some shops. The pirate museum was really fun, and we also went to the Pompey Museum of Slavery and Emancipation, which was very informative and nicely done. We walked through the Straw Market, but we're not really interested in knick-knacks, so that was mostly just to see what the Straw Market is all about. If you're uncomfortable with people trying to sell you stuff, skip that. My sister recommended the Main Street Ice Cream Parlor, which is near the other two museums I mentioned, but we weren't in the mood for ice cream while we were there. Of note, I paid for the $10 international day pass with AT&T, but struggled to get service in the shopping district, so you may want to have to destinations mapped out ahead of time.
Like most urban areas, keep an eye out and be aware of your environment, and you should be just fine.