No parts of Budapest are 'off-limits' to visitors, although some locals now avoid Margaret Island after dark, and both residents and visitors give the dodgier parts of the 8th and 9th districts a wide berth. Pickpocketing is most common in crowded places, especially those places where there are plenty of tourists.
Scams involving attractive young women known as consume girls, gullible guys, expensive drinks in nightclubs and a frog-marching to the nearest ATM by gorillas in residence are old news by now. If it seems too good to be true, it is. Caveat emptor - there's not much the police can do to help.
Waiters might try to pull a swifty over a foreigner, so don't accept anything you didn't order and check your bill carefully. If there's a discrepancy of only 15% or 20%, you can settle the matter easily enough by not leaving a tip. Otherwise, call for the manager. Finally, taxi rides can sometimes be unpleasant. It's best to call for one rather than hailing one off the street.