Multi Class + Negative Class Image Classification Strategies
There is no general strategy to anything in machine learning. Moreover, research is scattered over multiple domains so it becomes harder to get your head around, "the best strategy for a case". You can solve a single problem in multiple ways. Coming back to the multi class plus a negative class classification issue, this is one of "many approaches" you can follow:

Hierarchical classification: This is still an open problem. As the name suggests you perform classification on multiple levels starting from a root to internal node to leaf nodes (similar to tree or DAG structure). For eg. Given a digits data set ( 0,1,2,3...9) mixed with some letters (a,b,c....z). On a higher level you need to first classify if its a digit or not i.e. meta-class and if its a digit then you perform lower level classification task to identify the right digit.

There are different variants depending on your knowledge of labels/ groups at different level of hierarchy. For eg. if you don't have a pre-defined meta-class then you might have to perform clustering of similar classes at the higher level before final classification. 

Also, you may want to tune the weights in your loss function to identify a particular class efficiently at any level of the classification.   

P.S. My knowledge is limited in neural nets context. Definitely there would be more strategies specific to them.