Beijing is generally a very safe city, with low crime rate and severe penalties for those who are caught committing crimes. However, petty theft of bicycles can be quite common (almost everyone has had a bike stolen once) and purse snatching may happen in crowded, tourist areas.
The most common risks are getting cheated out of your money through scams or price gauging. The best way to avoid scams is to refuse all unsolicited services, including taxis, suspicous offers to show you artwork/tea ceremonies/etc. by strangers, and anything else that seems out of the ordinary.
There are no specific dangers to women in the city and women can move around freely without hassle.
Beijing traffic may pose the most dangerous risk of all as drivers do not always give pedestrians the right-of-way. This also applies to bicyclists who jostle for road space (although Beijing has dedicated bike lanes in most areas now).
In terms of health risks, there are some risks that may be unique to Beijing. Although improved, Beijing’s pollution levels can be very high during certain seasons and cause or aggravate respiratory problems. You can wear face masks or opt to be outside as little as possible during the worst days. In terms of food, exercise caution when eating street food and do not drink the tap water. Always wash your fruits and vegetables very well.