As the matter of fact, I think not only Dostoyevsky, but most of the other big classics of pre-20th century wouldn’t be recognized.
First of all, popular genres now are; fantasy, young adult, post-apocalypses, thriller, biography and others, have nothing to do with most of the classics written. “Don Quixote” for example, doesn’t fall in neither of those genres and obviously wouldn’t get as much recognition if he had been published now, simply because only small fraction of people read this genre today.
Secondly, it feels (and maybe I’m wrong) that people now, prefer more fast-paced and filled with action books. I very often hear and read complains how certain books like “Crime and Punishment” or “Anna Karenina” are plain boring, and there is nothing going in them. As I see it, their is a tendency to value real physical dilemmas more then psychological ones.
And lastly, taking the risk to sound like an old man, it seems that less teenagers and adults become interested in reading classics. If so many people are already refuse to read some big classic, there wouldn’t be a chance for it if it was published today.
As I conclude my sad observation, I want to point out that it is only my opinion and I would be more than happy if anyone could prove me wrong.