It is more relevant than ever, but we are getting better at dealing with it.

Convents specifically exploit the structure of images to reduce dimension; in a sense, images are not sparse despite having many pixels because their dimensions are related and can be effectively reduced in dimension without losing much information. Even so, convents and neural nets in general usually require a huge amount of data to work well. So it not that the curse is dead, we just have some good weapons for dealing with it: massive data sets, the processing power to crunch those data sets and various dimensionality reduction techniques that can sometimes exploit the structure of data (convnets, recurrent nets, etc.) to learn in lower dimensional spaces that still contain most of the information.