Hotel Captain Cook in downtown Anchorage is excellent. There are also a number of chain hotels which are also fine and would qualify as safe.(Obviously when travelling, it always pays to be aware of your surroundings, and use normal caution in even the safest surroundings). There is a lot to see in the Anchorage area, among them: the new Alaskan Native Heritage Museum which we visited last year. If you have time, rent a car and drive down to Portage Glacier and go to the Visitors Center. There is a large seaplane facility and nearby is the Earthquake Park which will give you a good understanding of the Good Friday Earthquake. Enjoy your visit to Anchorage and Alaska. Would go back in a heartbeat.
I had been referred to Captain Cook's (apparently the best & also priciest hotel in town) but no rooms were available. I stayed for 1/2 day (while waiting for plane back to east coast) at the Westmark Hotel which is also on 5th a block up from Captain Cook's. Across the street I had a late lunch at 'The Brewery'. There was nothing to fear in that area and I was traveling solo. I just returned from my cruise there two days ago. If you have never been, then I only hope you experience the feelings I have yet to define. It was truly an incredible journey. Please feel free to email me if you have any specific questions. Otherwise, be assured the wait will be well worth it.