I thought this to be a public fountain, but have in 2014 been informed it is a place for prayer, of a type called namazgah. I indeed found an article referring to these constructions, in Turkish, from which I understood that in Istanbul 153 namazgah's existed. The article also stresses the "kaynak" element, which is a source. So: one part was used for ablutions and as a fountain, but one might climb upstairs and pray.

This particular one is called Esma Sultan Namazgâhı (1871). The article has some pictures too.