When my wife and I visited Budapest 2 weeks ago, we wanted to go to the Szechenyi Baths because we heard that they were the best: classic building, outdoor pools, etc. After standing in line for an hour (we got there at 3 pm), we paid for admission to the pools with a locker. I asked the woman about towels, and she said, "They're downstairs." Silly me, I thought the towel was included in the admission price I just paid. When I got out of the pools and headed to the showers, I found that, no, the towels are not included, but they are available for a fee. There is a station between the womens and mens locker rooms where a woman rents towels for 300 Ft, with a 1,000 Ft deposit. Now, 300 Ft works out to about $1.30 US, but I was not walking around with 1,300 Ft in my bathing suit. After I walked back to my locker, got a 2,000 Ft note, walked back to the towel rental and got a towel, I was able to shower, get dressed and return the towel for my deposit. My wife was not so lucky. She wasn't carrying any cash and the woman would not accept her credit card. Since I couldn't enter the womens locker room and give her some cash, she ended up just rinsing off and air-drying as much as she could. We had a great time in the pools - just be aware of this hassle if you don't bring your own towel. I wish the woman who sold us our admission would have told us that towels were available for an extra charge.
I agree, the process is slightly convoluted at the baths. On my short vacation before I broke my ankle, I was able to make it to Gellert Baths, where I stupidly only brought enough cash for entry, and not enough for a deposit on a towel, as I left my jewelry and wallet in my hotel safe. It rained the day I went to Gellert, so my towel would have been pretty useless sitting out on a chair. I dried off in the sauna instead!
The deposit at Gellert for a towel is 4000 forint, so I should have gone back to Szechenyi instead. It's more fun anyway!