What occupations and fields are stagnating or decreasing in growth?

I don't really think occupations are stagnating, I think many are capable of being replaced. In the coming years, artificial intelligence and automation can and will take up many jobs. I do think, however, that occupations that involve human connection/bonding could not be replaced. Or jobs where you're meant to inspire people or do some storytelling, also couldn't be replaced by AI. But that's just my two cents :)

http://www.boston.com/jobs/untagged/2013/12/30/30-of-the-fastest-declining-occupations

A good pressure test to put on any industry or job is to think about the effects that technology and globalization have had on them. For instance, is there a digital process or a software service that can do the same thing a human needed to do? Can the whole process be taken online when there used to be multiple humans involved in the transaction before. Can the internet be used to do the same job for cheaper overseas? Many technical types of positions that used to require someone to manually read something or process a transaction, can now be done online. There are also a lot of fields where barriers to entry are very low, so it's becoming increasingly harder to make money off it... like journalism, graphic design, etc. That's not to say there aren't still very successful people in these fields, but technology and especially free software has made crowd sourced design, copywriting, starting a blog, translating language, finding information, etc so much easier, so the person who used to do that as a job can have a harder time making money from these specific skills.