Two issues: one is that the temptation will exist to wring out a wet towel after use:
IF ONE BATHES IN THE [heated] WATER OF A PIT OR IN THE [hot spring] WATER OF TIBERIAS AND DRIES HIMSELF EVEN WITH TEN TOWELS, HE MUST NOT FETCH THEM IN HIS HAND. BUT TEN MEN MAY DRY THEIR FACES, HANDS, AND FEET ON ONE TOWEL AND FETCH IT IN THEIR HANDS
One person alone cannot carry towels, even if he uses many which are each not very wet, because he will be tempted to wring it. But many if even they use few towels and thus it will be very wet, will remind each other not to!
And issue two is carrying towels in a public area on Shabbat.
R. Simeon said: One may dry himself with one towel and bring it home.
Rab Judah said in Samuel's name: A person may dry himself with a towel and carry it home [wrapped round] his hand
That is, the towel needs to be worn like a garment, not carried as a burden. And just to be clear:
R. Hiyya b. Abba said in R. Johanan's name: The bath attendants may bring women's bathing clothes to the baths, providing that they cover their heads and the greater part of their bodies in them
There is a related ruling regarding the responsibility of billeting foreign soldiers – which often involved doing menial labor for them – even on Shabbat:
Raba said to the citizens of Mahoza: When you carry the apparel of the [billeted] troops, let them drop below your shoulders.