What's the safest thing to do with your passport and other valuables when you go to the baths in Budapest? Leave them in a hotel safe? It seems odd not to have the passport with me at all times--I always wear a money belt.
You can rent a locker for your personal belongings while at the bath. If you want to take your passport, just leave it there with the rest of your belongings.
Hungary is one of those countries where it is the law that you carry your passport at all times. Assuming you are law abiding, just put it in the locker at the bath house. I wouldn't carry too many valuables with you when you go, but only for convenience. If your accommodations are in a reasonable location for tourism you can reach most of the bath houses in under 10 minutes on the public transport; so go to the room, drop the non-essentials and go have fun.
Rent a locker and put it there.
Make sure you try the non-tourist bath in the City Park north of Heroes Square. Szechenyi Furdos, or something close to that. It has its own Yellow Line subway stop. the entrance is opposite the carnival entrance. This place is worth the trip all by itself. Take your own towel even if 'borrowed' from the hotel.
Széchenyi is probably the most touristy of the bath houses, but that really doesn't matter. I am going to guess that more than half of the visitors on any day are locals and being among locals is what a lot of folks value when exploring new places. If you are the least bit squeamish about wall to wall butt to butt naked bodies in a locker-room, then rent a Cabana which is really just a glorified closet with a lock. The bath house is identified as Széchenyi fürdő on the M1 Metro Line which is the station immediately after the Hősök tere (Hero's Square) station. All metro line directions are identified by the end station. In this case you want to get on the Metro towards Mexikói út.
More information at www.szechenyibath.hu/
I left my money belt with passport in the hotel safe, although I rented a cabana closet. Have fun!
I left my passport in my rented apartment but brought a photocopy with me and left that, along with my clothes and a bit of cash in the cabana (Szechenyi) or locker (Veli Bej and Gellert).
I did have a bit of a moment when, on the way to Veli Bej, we encountered blocked off streets and riot police and didn't know what was going on. At that point, I was thinking maybe I should have my actual passport on me. However, it turned out to be for the Gay Pride parade.
If I did carry my passport, then I would just lock it in the cabana or locker, and maybe have it in a Ziploc bag, just in case it got dripped on.
Hungarian law requires you to carry your passport at all times. I would sort of expect that the bath house locker qualifies, but not the hotel safe. Same law exists in most of the old Eastern Block countries, and Belgium.
Dang, I was breaking the law in Hungary and never knew it.
Besides, who is going to know or actually catch you? And if they dont know or catch you, well, you arent really doing anything wrong. And besides, YOU ARE AN AMERICAN! And who are they to have unrasonable expectations.
Even more inconvenient is citizens have to carry their ID cards too. But they arent Americans!
Personally i rarely carry my passport, but i know i am doing wrong. And i have a local support network to bail me out