There is some 80's music that even I don't understand why it was popular.
A certain Boy named George (and another George named Michael) are proof of this.
Boy George was popular because he sung the type of music that all the teen girls were into at the time. I'm pretty sure if they understood cross dressing wasn't just a gimmick for him that they wouldn't have been as enthusiastic.
George Michael (with Madonna) fundamentally changed how we talk about sex in the US. His music seems quaint now, but it could be argued that without him, Katy Perry wouldn't be singing about seeing Peacocks. I have a good deal of respect for him...
But if I were to choose, I would say that Depeche Mode had the most influence on modern music of anyone from the 80's. Before they came around, using electronic keyboards and computers to make music wasn't treated seriously.
That's all in my highly bias opinion of course...