As the background, I am a graduate student in high energy physics, mainly theoretical physics. Recently, I have sen that there are a few scholars who don't put the preprint versions of the paper in the preprint servers such as arxiv. I must add the fact that for many of the journals the self-archiving of the preprint is allowed, and I see no point for not putting the preprint version submitted to peer review on arxiv. I have seen that there are some old researchers and also some not so old ones who don't put the preprint versions of the paper on arxiv and sometimes make it difficult for some papers to be read. Hence, the question would be on how to promote open access and convince people to be opened within this direction.
From what I see, some organizations are solving this by making self-archiving mandatory. In the UK, research funding bodies are now starting to use only open-access publications to assess the quality of research. As a result, the universities are starting to require their academic staff to self-archive publications. Also, I've seen a conference organizers discussing whether they should require the authors to self-archive the accepted papers.
This is a very broad question that the open access (OA) community has been discussing for ages and, as you point out, is not solved yet.
Universities have tried adopting open access policies to encourage researchers to share their works. (See ROARMAP for an overview of these policies.) However, many of these OA policies require researchers to deposit papers in institutional repositories (such as DASH for Harvard) instead of topic repositories such as the arXiv, for various reasons.
Many research funders also encourage OA (for instance, the Wellcome Trust or the BMGF), but they often foster hybrid OA: paying publishers to make the article freely available even if it was published in a closed journal. These fees are called Article Processing Charges (APC). These policies do not focus on fostering self-archiving by authors.
Topic repositories are successful when they are adopted by scientific communities: this is the case for arXiv (originally in physics) but also for more confidential platforms such as IACR eprints (cryptography) or LingBuzz (linguistics). By definition, scientific communities span accross universities and funders, so these repositories are quite orthogonal to institutional efforts towards OA. I think the best way to foster them is to have some influence in your community and advertise the repository you like.
Note: I am involved in the dissemin project, which tries to help researchers upload missing papers to open repositories (not specifically topic repositories though). Feel free to join or to send us your comments.