My WebSocket client DandelionWebSockets.jl is now released with v0.2.0, with Julia 0.7 and 1.0 compatibility.
Some of the code was re-written and improved. The handshake code, which makes the connection, is still a work-in-progress, but appears to work somewhat well. For instance, it can make a TLS connection to Slacks WebSocket API. However, YMMV for other WebSocket APIs.