Please do not worry. Sevilla in general is very safe, clean and tourist-oriented. El barrio Santa Cruz is very safe in particular. It is the ancient quarter and is replete with history. It is well taken care of and full of tourists. It is also a very chic neighborhood with the highest rents and property value in Sevilla city.
There is the occasional gypsy who will try to persuade you to give them coins in exchange for a flower or rosemary. They are congregated mostly around the Giralda cathedral.Ignore them and keep on walking. Remember that guns are illegal in Spain (as in the rest of europe) and violent crimes are still virtually unheard of.
I feel by far safer walking through urban areas in Spain than those in North America.
The only neighborhood that may be dangerous is tres mil viviendas. Even so, I would rather walk through here in the dead of night than walk through the projects in NYC or Chicago in plain daylight. This is low-income property built by the government. It is on the ourskirts and I really doubt you will see or even hear of it during your stay. Mostly gypsies or immigrants live here as the rent is next to nothing.
Sevilla is veeeery safe! Have a beautiful time!!!