How to encourage self-archiving of preprints in science?
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.