Most of the Fredericksburg area is "very low" on the crime risk chart. One of the zipcoded regions is "average" risk. Most of the crimes are non-violent (thefts).
There are sixty-three registered sex offenders in the area (see website below).
There is nothing in the crime statistics to indicate any one thing to worry about in being safe in Fredericksburg. Just use common sense and the same methods you use at home for personal safety and you should have a fine time here.
Central Park is a massive shopping complex west of I-95 and north of Rte. 3 built by local millionaire Carl D. Silver. This ugly place replaced a golf course (which in turn replaced a forest) and is one of my least favorite places in Fredericksburg. Central Park is complete with Wal-Mart, Target, Lowes, etc; best to stay away if you can.
In most places, parking at the weekend is easier. However, when you have an historical place noted for numerous antique shops and little area in which to park, parking is a full-time problem. There are some public car parks, but they fill up quickly, so bring along your hiking shoes as you may have to park blocks from where you want to sightsee, dine, or shop.
Be careful if you do decide to venture along the river. While it can be a fun place, if you are not careful, there are ample opportunities to hurt yourself, and that's not even getting toward the water. The area across from the church that I have mentioned has a lot of rocks and a small trail to get to the river, but it is narrow, and the walking between rocks and brush isn't the easiest. (I'd like to say it's totally unspoiled, but it's not, due to people who don't pick up after themselves, and due to those who decided to tag some of the rocks to remind others of their unfortunate eternal presence.) But it's not all like that, and I enjoy being there, listening to the water, photographing the river and viewing the different scenery around there.