"Fewer responsibilities" is correct. The sentence you read is wrong. If you use 'responsibility', it is considered a quality and is therefore uncountable. "This job has less responsibility." "Responsibilities", however, is considered countable, because it is assumed that you could list them individually. "This job has fewer responsibilities."

I have SIX LESS EGGS than I had before, because I made an omelet. (12 - 6 = 6) I have TWO LESS RESPONSIBILITIES than I had before, because the brother and sister went home with their mother. (8 - 2 = 6) But the intended meaning of "less" here is similar to "subtract" or "minus" as you would use in a mathematical operation. On a grammar test where they are testing your knowledge of count and noncount nouns, choose "fewer responsibilities" for the reasons FDMAXEY explained above.