Fort Worth is a safe place to visit, if you use common sense.  Almost any area is safe during the daylight hours.  Just be aware of your surroundings--look out for people wandering around, don't leave valuables in plain view in your car, etc. 

After nightfall, most areas you will visit are safe.  The downtown and Sundance Square area is well-lit and usually have several policemen on foot that patrol.   Camp Bowie Blvd and the Cultural District should also be safe at night.  You may occassionally run across someone asking for money for gas--this is almost always a scam, even if they look presentable.  It is best to avoid a few areas of Fort Worth that have more crime after dark, including most neighborhoods in east/southeast Fort Worth (the area south of I-30, east of I-35, and west of Loop 820).  However, this area doesn't really have many tourist attractions--you'd only end up there if you were lost.