Thats true that a lot of uzbeks but also armenian women comes to batumi in high season. Gonio is a village between batumi and sarpi(border) on blacksea shore but developing fast as an holiday resort, mostly Turks invest there for Hotels etc...
Government wants to make Batumi an all entertainment center, sea, hotels, casinos, women, clubs, concerts etc.. So there is almost no anti-prostitution measures.
Anyway, nice places, I recommend to Turkish friends to visit there, friendly people, beautiful nature, cheap prices, even no passport required on the border, your ID card is more than enough.
There are from my camera ,
Sarp Border Gate
Gonio, early in the morning
Batumi Turkish mosque