They're both right, but they mean different things. It all comes down to grouping: "I'm busy helping [Japanese people] [learn English]" - I'm busy teaching Japanese people English. "Japanese people" is the accusative and "to learn English" is the dative. "I'm busy helping [Japanese people learning English]" - I'm busy helping people from Japan who are learning English. "Japanese people learning English" is the accusative. Basically, the infinitive is not a tense! It is a separate grammatical construct called a "mood" (in most other Indo-European languages it's contrasted with the "indicative" (something that is happening) and the "subjunctive (a possibility), and occasionally the "optative" (something that might happen but probably not).)

"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

Thank you for your answers, but I'm curious as to *why* the first sentence would be considered grammatically correct. Recently I got asked to explain the difference between the two sentences, English comes to me instinctively so I've never really thought about it. I was hoping someone could highlight how they are different according to grammar rules.

Hi, I would use „to learn”. I've been busy (with) helping Japanese people to learn English.

Of the two examples, the first one is grammatically correct. You could also consider eliminating 'with' from the sentence. It depends on the conversation. "I have been busy helping Japanese people learn English."