The "postprint" is the version accepted by the peer-review process of a journal, often after some revision. If you transferred copyright to your publisher, then postprint archiving requires the journal's permission. However, many journals --about 59%-- have already consented in advance to postprint archiving by authors. Some will consent when asked. Some will not consent. For publisher policies about copyright and author archiving, see the searchable database maintained by Project SHERPA.
If you have not yet transferred copyright to a publisher, then ask to retain copyright. If the journal does not let you retain copyright, then ask at least for the right of postprint archiving. If it does not let you retain the right to archive your postprint, then ask for permission to put the postprint on your personal web site. For many journals, the difference between OA through an archive and OA through a personal web site is significant.
The chief benefit of postprint archiving is reaching a much larger audience than you could reach with any priced publication (in print or online). Reaching a larger audience increases your impact, including your citation count. Many studies confirm that OA articles are cited significantly more often (on the order of 50-300% more often) than non-OA articles from the same journal and year.
Because most non-OA journals permit postprint archiving, it is compatible with publishing in a non-OA journal. Don't assume that publishing in a conventional or non-OA journal forecloses the possibility of providing OA to your own work. Depositing your postprint in an OA repository takes, on average, 6-10 minutes. You've already spent hours trying to get your work in front of the audience that can use it, build on it, apply it, cite it. The last few minutes can vastly amplify that effort.
Faculty needn't donate their time and labor to journals that lock up their content behind access barriers where it is less useful to the profession. Universities should support faculty who make this otherwise career-jeopardizing decision. Faculty don't need to boycott priced journals, but they don't need to assist them either. Ask the journals where you have some influence (as editor, referee, or author) to do more to support OA. Start an in-house discussion about converting to OA, experimenting with OA, letting authors retain copyright, abolishing the Ingelfinger rule, or declaring independence (quitting and launching an OA journal to serve the same research niche).
Work with your professional societies to make sure they understand OA. Persuade the organization to make its own journals OA, endorse OA for other journals in the field, and support OA eprint archiving by all scholars in the field. Ask the societies where you pay dues to consider these actions. Ask other members to help you change access policies at the society.
Volunteer to serve on your university's committee to evaluate faculty for promotion and tenure. Make sure the committee is using criteria that, at the very least, do not penalize faculty for publishing in peer-reviewed OA journals. At best, adjust the criteria to give faculty an incentive to provide OA to their peer-reviewed research articles and preprints, either through OA journals or OA archives. Work with your administration to adopt university-wide policies that promote OA.
Make sure that new researchers (and experienced older researchers too!) understand their self-interest in OA. Make sure they understand that OA increases the impact of research articles. Or, at a minimum, don't let myths about OA circulate without challenge. When you meet students, colleagues, or administrators who are curious and want to know more, or who misunderstand and need some facts, direct them to this guide.