It is the responsibility of the employer. The "pemohon" applicant for the work permit (RPTKA) and for the KITAS is the employer. The applicant is the one responsible to pay.
However, is there any enforcement of this? No, never heard of it happening. And Manpower and Immigration aren't going to get involved, they just need to be paid the appropriate fees and don't care the source.
If the person already has a contract for employment, they could try to get it enforced through the courts, but we all know that will cost far too much money and headache to possibly still go nowhere.
His only real option is to negotiate with the employer and make explicit agreement his pay, benefits, and responsibility for work permit and visa fees. Of course this should have been done prior to moving, but he's past that point now, leaving the employer in the stronger position to negotiate it now.