Yes, it is completely secure. For the browser extensions, we use WSS (secure WebSockets) to connect to our remote machines, and transmit your data. WSS uses SSL over port 443, and therefore the data is encrypted during transmission. For the binaries, we again use SSL to ensure that your data is transmitted securely.
In addition, if you are testing from behind a firewall, BrowserStack testing doesn’t require you to create special rules for the connection. We use standard ports, which are accepted by firewalls universally. Once you set up a Local Testing connection, only you can access your files in your current remote browser. Read more on security.