Selection of base model for transfer learning
There is no specific rule associated with the base model selection for transfer learning. It is generally a trade-off between model precision and resource allocation. As the number of layers increases, the number of parameters increases and the model becomes more and more resource heavy but deeper models tend to have better accuracy over shallower counterparts. Here's a comparison:

 

Apart from that also refer to this question: Which is the fastest image pre-trained model? if the speed of the model matters.