What is this game that is about connecting streams?
You possibly played Ta Yü.

It's one of many games that follow the "connect two sides"-pattern, more popular ones are Hex or Quoridor, versions can even be played on a Goban (Go board)

However, there are differences to your Description:


Players are starting in the middle of the board and try connecting to two opposite sides
The "streams" are apparently never colored red: