DOIs are assigned by CrossRef on behalf of members. CrossRef is an association of scholarly publishers, so the members are therefore publishers rather than authors.
DOIs were introduce because of "link-rot": citing a paper by URL and within 6 months the URL doesn't work. When a publisher joins CrossRef they make a commitment to keep the DOI link metadata updated, so the DOI always points to the paper if, for example, the publisher is bought out and all the URLs change. This agreement is an important part of the DOI system and explains why publishers are members rather than authors.
Assigning a DOI is normally done during the process of publication, and will normally be done automatically. That's not to say that DOIs can't be assigned after publication: DOI 10.1098/rstl.1672.0051 is for a paper published in 1672. Publishers generally assign DOIs to new content but not to back files because new content is more likely to be cited.
It's not free to register a DOI, off the top of my head, around a dollar per DOI. It's normal (and cheaper) for publishers to deposit back-files, i.e. for publications older than 2 years (again, off the top of my head).
So, if you want to get a DOI assigned you can go to the publisher and request that they register a CrossRef DOI.
STM (Science Technology and Medicine) is a very fast-moving field with lots of papers published and cited. Humanities publishing tends to move at a different pace and tends to have different priorities. Therefore you may find STM publishers more likely to assign (and to need to assign) DOIs than in other fields.
The alternative is that you get a DOI from another organisation, for example FigShare. There are examples of people registering FigShare DOIs for their publications.
CrossRef was created to solve the problems of assigning persistent identifiers (DOIs) and linking between papers (one DOI cites another DOI). CrossRef also has lots of other infrastructure, such as bibliographic metadata, FundRef, CrossMark, metadata API etc so it's the best place to register scholarly publications.
Disclaimer: I work for CrossRef