If RealPlayer is isntalled on the user machine, every time the video is played, the button will show up sugegsting the user to download video to a local drive.
How can we supress this button so that when the video is played on our web page, the user won't see it?
As far as I know, you can't. That was a dark, dirty path that Real decided to go down a couple years ago. It essentially gives users a direct tool to copy potentially copyright content. Granted, tools to do exactly this with Real content has existed for many years, but now it's dirt-simple.
You may want to consider switching your media to be Flash-based. The Flash Media Server (from Adobe) can encrypt and obfuscate the data stream, making it more difficult to get a clean copy of the video.
Presumably the user installed RealPlayer in order to have this capability. Why do you think you should be able to alter the normal functioning of software on the user's machine?
Use something other than the abysmally terrible RealPlayer plugin, whose install base is in any case dwindling fast.
Hopefully, Flash streaming and HTML5 video should together purge the web of the bane that was proprietary video plugins. WMP and QT are bad enough, but Real was vile.