Buses are safe, but use your common sense, and put your things in internal pockets, just to be 100% sure. Taking taxis from areas frequented by foreign tourists may also pose a threat as some of these taxis may be operated by con men waiting for an unsuspecting victim. This is especially true for taxis around Gara de Nord where their associates actively try to lure you into such cars. If possible, avoid taking cabs from Gara de Nord unless you are familiar with the taxi operators there.
One rule of thumb is to go with older taxi drivers, since they will be more cautious and only try to get a bit extra out of you if they scam you, unlike young drivers who will claim a trip costs 3-5 times as much as it should, may claim the meter does not work, and may try intimidation tactics to make you pay.
As strange as it sounds, you'll see that Bucharest is a far safer city than its western European counterparts. Statistically Bucharest is one of the safest capitals in Europe, far safer than cities like Berlin, London, Rome, etc.
Bucharest has perhaps the largest population of stray dogs for a city in eastern Europe. Although their numbers are gradually decreasing due to projects by the City Hall, they still remain a threat to safety and, at night, they tend to form packs which greatly increases their danger. Rabies vaccinations are recommended but not required, as there have been no rabies cases in Bucharest since 1979. Most dogs will not give you a problem unless you go out of your way to pester them, but many dogs have been treated poorly. Be extremely wary of them, and do not approach a stray dog if you are alone. It is perhaps best to walk around in a group or walk where you see other people. As of 2009, stray dogs are an ever more rare sighting in the city center.
Like most other big cities, walking around at night isn't safe in some parts of the city like Pantelimon, Ferentari, Giulesti, and the Gara de Nord area. If you must travel into these neighbourhoods, it's safer to take a taxi.
Gara de Nord is not particularly dangerous to walk in, but avoid suspicious-looking characters, and if you feel that you are being followed, just walk into the station. Gara de Nord and its surroundings are populated by homeless people and children. Be careful, as many street children use an inhalant drug (equivalent to huffing paint) and may be dangerous. As heartbreaking as this problem is, it's best to avoid any contact. If you do wish to give them something, buy food for them, don't give them money. Ferentari is a gypsy enclave in Bucharest and, while not as dangerous as it used to be, it's not advisable to walk there at night. In fact it is better to avoid it completely. For the traveler, there is nothing of interest there (aside from what Bucharest sounds like in foreign newspaper articles) so you should have no reason to go there to begin with.
The unofficial red light district is Mătăsari, which is also a popular place for clubbers and pubs; you can walk there without any worries because it's always crowded and lively, but avoid talking to strangers in that particular area, especially Gypsies. As of 2009 there have been a lot of crackdowns on pimps and prostitutes in the Matasari area, so be careful or you might wind up spending a night in jail and with a hefty fine if caught soliciting.
In the event that you do get caught in a police raid, do not attempt to bribe your way out of it with so many of them around as you might get into serious trouble. Police are more inclined to take bribes from locals than from foreigners so do not contribute to this phenomenon that has been plaguing this country for so many years. Police corruption has been vigorously fought in the past years, and it is not as generalized as it used to be in the 1990s. It's always better to walk on boulevards and avoid alleys and backstreets.
The crime rate is low, but a traveler must always be cautious. Violent attacks are very low, but if attacked just yell, "Ajutor!". It is very difficult for anyone to get away with violent crime because as everything is packed so closely together, any loud noise will attract attention. This is truly a city that doesn't sleep. You'll find people out and around at all hours in most parts of the city. Police men are pretty friendly and the younger ones speak English, so you can ask directions. In the event that you do need to report a crime to the police, do not hesitate and proceed to the nearest police station. They will often help you to the best of their ability.
One must be incredibly careful as a pedestrian in Bucharest. Drivers are inconsiderate and often do not obey traffic signals. NEVER assume a car will stop for you at a red light--be vigilant at all times. This is definitely the biggest hazard in Bucharest, not so much in the daytime, when crowded streets make it impossible to drive cars at high speeds, but, at night, the streets clear out, a lot of illegal races taking place with reckless driving on main boulevards.
Asian tourists are more likely to be seen as an easy mark for dishonest taxi drivers and other criminals. It does not make a difference if you are Asian-American or are from Asia. Some young Asian women may also get a lot of perverted looks from men all around the city - be prepared to be stared at especially if you are traveling alone, though some men will stare no matter what.