I did a tour to Egypt including the Nile River cruise but not with Viking. I can comment though on the areas where the ships dock. The areas felt safe to me when I was at the ports during the day. However, Egyptians do not run along a street. You will really stand out as an oddity. Also, some areas were crowded and not sure how you could navigate around everyone enjoying their day along the River as well as vendors.
You out didn't mention if you are male or female. Running attire for ladies definitely NOT appropriate in Egypt, even touristy places like Luxor and Aswan.
All the men I remember seeing wear slacks and shirts, no atheletic wear.
Hope this helps!