Budapest is a very safe city, although it had a reputation for scams and car theft some decades ago. This, along with the notorious fraudulent taxi drivers, is mostly a thing of the past though since the city was tough on crime lately.
You should still be aware of pickpockets when near train stations, in highly frequented shopping streets and around tourist attractions. The same goes for women in the inner city telling you their story of being new to the city and searching for this one great bar and asking if you could walk them there. There is a high chance of this bar having a sinister bouncer that demands some money from you while the girl gets back to the city being all new and lost again.
8th district and parts of 9th district are considered less safe then the rest of Budapest in general.