One way of checking this would be to run a packet sniffing tool such as wireshark, as this can give you information about the protocols being used, and can show you similar information to what someone else would see if they're sniffing your connection.
I'm presuming here that Battle.NET uses HTTP(S) for authentication so essentially you'd be looking for the difference between a plaintext HTTP connection and an HTTP(S) connection.
In practical terms a good way to do this is to close as many other programs before you run the sniffer, to minimize noise and make it easier to isolate the relevant traffic. If you can identify what IP addresses the servers run on you can set-up a capture filter on wireshark for only those IP addresses, which would make it even easier.
Then start the sniffer and log-in to Battle.NET. Once you're logged in stop the sniffer and examine the output.
what you're looking for is the protocol that's in use for the connection if you see connections using HTTP and you can see the contents of the connection (right-clicking one of the packets that's sent to the server in wireshark and using the "follow TCP Stream" should show you some details.)
If the connection is using HTTP(S) then you shouldn't see any plain text after the connection is established, whereas if it's using HTTP you'll see plain text detail of the traffic being sent back and forth.
Another quick and dirty check would be to save the packet capture file from your login and use a string search tool (like findstr in windows or grep in linux) to check for your password or username. If you see them then they're being sent in the clear, if you don't then they're at least being obfuscated, and may be being encrypted.
There's still some other risks, especially if you're using someone elses computer in an internet cafe as they could install keyloggers if they wanted to and/or modify trusted certificate authorities which could allow them to intercept traffic sent over SSL.
Bottom line is that if you use a system controlled by someone else on their network it's likely that they can get access to data you process at that time.