Unclear. Control probably goes too far – the Braavosi are pretty heavily invested in their maritime navy as opposed to a riverrine navy, and trying to control the Rhoyne would mean persistent conflict with Norvos, Qohor, and Volantis, on terms that would be pretty bad for the Braavosi. It would be a huge investment with a poor rate of return, when maritime control offers much better profit margins. 

Rather, I think the Braavosi are only interested in access to the Rhoyne. It gives them a much larger market they can sell into and buy from, it speeds up traffic for longer voyages, it offers them substantial military advantages while risking little of their own since the Sweetwater doesn’t link with the Rhoyne, etc.