At the end of Siman 302, the Shulhan Aruch prohibits drying drinking glasses with a towel on Shabbat, out of concern that the residual water in the glass will lead to "Sehita"-wringing out the water from the towel. The Poskim question whether such a small amount of water is likely to lead to "Sehita." Therefore, the Radbaz (Rabbi David Ben Zimra, Egypt-Tsfat , 1479-1573), as cited by the Mishna Berura, disagrees with this ruling. Hacham Bension concurs, saying that Maran’s ruling only applies to thin cloths. However, today, our towels are thick and do not present a problem. The only concern may be when drying a very narrow glass, which requires forcing the tip of the towel into the opening.
It is permitted to dry glasses and other utensils with a towel on Shabbat.