A-list movie stars of blockbusters that have been filmed in Hungary talk about Budapest as a place where they felt totally safe. Stars like Tony Curtis, Angelina Jolie or Tom Hanks described the city as "walkable", "livable" and a "totally safe city even at night". Harrison Ford rode a bicycle alone through tiny remote villages, stopping every now and then by small local pubs.
Drug-related crimes are less frequent in Hungary than in many other European countries. The possession of firearms is subject to strict authorization, and is not common at all. Regarding natural disasters, Hungary is also one of the lowest risk countries. Female tourists can also feel totally safe, even if they travel alone.
However, there are a few things you should be aware of, as you would probably do anyway. Most of all: use common sense. You may encounter pickpockets at crowded tourist destinations or public transport. So, as always, take care of your personal belongings and documents. Order cabs by phone or mobile apps, instead of hailing them on the street. If you can, avoid restaurants the taxi driver recommends to you: the drivers often get commission after each guest. "Three card trick" scammers have not entirely disappeared from the streets either. Remember, this is a scam. There is no way you can win.
Check the price list at bars and restaurants before ordering. Like at any other place in the world, prices may be higher at major tourist areas than at places visited primarily by locals.
The nightlife of Budapest is remarkably vivid and exciting. Simply use your common sense at bars and clubs, and check the bill if necessary.