SFTP is another protocol. SFTP is built over SSH (precisely, over the transport protocol used also by SSH). You might think about FTPS, which is FTP-with-SSL. However, there is no easy (or even hard) way to mix HTTP and FTP traffic over the same SSL session.
If you went to the trouble of establishing a secure, authenticated HTTPS session, then my advice would be to use it, and provide a Web-based upload system. Uploading single files is easy enough. For bulk uploads, you will want a richer user interface (to select many files, upload directories recursively,...) and, for that, you will need some code on the client side (not Javascript, though, but code which has full access to the local files). It is kind of tricky to reuse an existing Web session (for instance, if the local code is a signed Java applet, then that applet must hand over the file contents to some Javascript code, which will then do the upload through the browser). Some basic Googling points to CuteUpload, which is an ActiveX control (thus limited to Windows+IE). Alternatively, distribute to your users some application (which can be a signed Java applet) which will open the connection to the server and do the authentication itself (instead of using the browser), at which point you do what you want. Whether this is applicable to you really depends on your exact context.