In general, Budapest is a safe city. I've lived here ten years and been out and about all times of day and night in all parts of the city without any problems. That said, sometimes things do happen, but usually the worst is some kind of pickpocketing. As long as you stay away from strip clubs and avoid overly-friendly strange ladies on Vaci utca who know a 'special' place where you can take them for a drink, you should be just fine. What are you looking to do while you're here? I'm happy to offer advice or suggestions..
Have Fun!