They have lockers and cabins that you can only open with a watch. Been going to baths for like 20 years, never had anything stolen from those (or know anyone who had). As for baths, the "big 3" are Gellért (nice architecture, you can also go outside, there's a sauna on top), Rudas (classic Turkish bath, just check when men's/women's/mixed days are beforehand) and Széchenyi (big outdoor area, also nice building). I like being outside in the warm water when it's cols outside, but checking out an original Turkish bath is also an experience, it's up to you really. You can find more baths and info about opening times, prices etc. on the Budapest Gyógyfürdői website.