I have a cassette which was prepared by mixing two tracks.
(Example - Mozart on the left track and Beatles on the right.)
If I simply disconnect one speaker, I can hear the other
channel's sound properly. I want to make an mp3 clip with
sound of only one channel.

Last year, I had made an mp3 clip off this cassette and a friend
had successfully separated the channels using Goldwave. But I tried
doing it and BOTH the channels of my mp3 clip carry BOTH
the tracks on the cassette (Mozart and Beatles). The only difference
is that last year I had used a sound card to record and this time
I am using a Yamaha chip on the motherboard. Could that be the
reason why I can't separate the channels?

I had written ...

The problem was solved by setting properties of the
'Yamaha DS-XG Audio Config' (from Control Panel).
After I set Direct Sound 3D to 'Speaker', there
was proper routing of the channels even while playing
the cassette and at all subsequent stages of recording,
editing the wave, etc.