How do I FTP a data in a MS Excel workbook on the user's desktop to a mainframe (dataset)?

First, export the data in a format such as CSV.
then you can download a tool like WinSCP and use that to transfer the file.
Once it's there, you'll have to put it in a place you want on the mainframe.

What is the OS on the mainframe?  

Unless the OS has a program that can read xls or xlsx formated file you need to save it as a CSV or text file and then use whatever file transfer method the mainframe supports.  Typically you would contact the people that support the mainframe to find out, but it depends on the OS.

IBM zSeries mainframes can run one of 4 operating system: z/OS, z/VM, z/VSE, and/or Linux.

Linux on a mainframe is just like Linux on any other hardware.

z/OS suppport FTP, FTPS, SFTP, SCP, RCP, and IND$FILE.

I don't deal with z/VM or z/VSE, but I would assume they support FTP, FTPS, SFTP, and IND$FILE.

The mainframe support staff may also give you some special command to issue if you are ftping to z/OS, z/VM, or z/VSE.  

Since you have two questions that are basically the same, you may want to press the "request attention" button and ask to have one of them closed.