Overall, Budapest is one of the safest cities in Europe. There are problems with pickpockets on crowded buses, trams, or metros. Caution should also be taken in crowded shops.
The U.S. government as well as others cautions travelers against patronizing certain bars and clubs. You can view the list here: http://hungary.usembassy.gov/tourist_advisory.html.
Due to some political issues, there have been demonstrations that have suddenly erupted during various times of the year. For the most part, they tend to localize in the park in front of the U.S. Embassy, in front of Parliament, and at Heroe's Square. In the past, there have not been any threats to tourists, if you just stay clear of those areas on the day of the problems. Most of them are short-lived, so will not disrupt your stay. Your accommodation should be able to give you a warning if something happens to occur during your stay.
Gay travelers should take care and caution during the time of the Gay Pride events, which are no longer specific dates due to problems in the past. They are now held at varied times each year. The conservative right has demonstrated and attacked marchers in the past. Otherwise, there have been no problems for tourists.
Women who are scantily clad may find men will give them the sexist wolf-whistles and comments, but other than that they are harmless. Until recently, men beyond 25 years old or so did not wear shorts in public during warm months, but this is slowly changing. Many men do not wear hats, but a cap is usually a sure sign of a tourist.
Drinking alcohol on the street, when not seated at a cafe, or on public transportation is illegal.