"I've been busy with helping Japanese people learning English"
It sounds like you are just generally helping Japanese people, but only ones that are learning English. "Japanese people learning English" is describing the subject as opposed to establishing an action, so this does not convey that you are helping them learn English. Instead of "helping" being applied to "learning", it's applied to "Japanese people learning English".
"I've been busy with helping Japanese people learn English" is broken up for a clearer understanding. "helping" is comfortably applied to "learn" to specify how you're helping, rather than the subject's action becoming a description.
Hopefully this helps, I might be horrible at this :P