If Mozart had been born in modern times would he be more and less successful than Kanye West?

Kanye West is hardly the most successful musician there is, he's just an outrageous self-promoter. Saying "I'm the most successful musician that ever lived" doesn't make him that.

Mozart would probably be more successful, assuming that we're translating his popular success from his time to ours. He was incredibly popular and a massive celebrity across Europe, especially as a child.

Mozart wouldn't be doing the same kind of music as Kanye West. It's like asking if American opera composer John Adams is more or less successful than Kanye West. As someone pointed out, Kanye is constantly seeking publicity and attention, and his "music" might appeal to a certain wider audience (though not every popular music fan necessarily cares for rap, and not all rap fans necessarily cares for Kanye West). However, John Adams can be considered successful in his genre.

Mozart would probably still be a musical genius. it's hard to say how he would react to the kind of media available these days. Would he put his latest composition on Youtube or release it on I-Tunes? Who knows? It's all wild speculation.

Mozart was a product of his time, and unfortunately Kanye is a product of ours.