This is a very difficult question. However, I think that we could attract more graduate students (actually, more people in general) following three lines of action. Many of these things have been already said in previous comments and answers by a number of people, however I think it may be useful to collect them here.
Clarity and guidance: it is important to improve as much as possible the customization of the modal window for the first-time askers. This is discussed very well here, but perhaps something more can be done.
Welcomingness and tolerance: in order to overcome the participation hurdle, we should make an effort and be patient with the new users, showing that we appreciate their good questions, even if they are not expressed in a perfect form. When possible, we should suggest to the more experienced members of the community the use of constructive comments and editing instead of downvoting. If we set an example, we encourage people by our behaviour to behave or act in a similar way.
Advertising: we must overcome years of "MO is intimidating", so it is important to make an effort and promote the site among our students and colleagues. We should point out that the community is working hard in order to build a professional and inclusive environment, and that nowadays MO is for everyone wanting to ask good questions and to learn new Mathematics.
I do not know if it is currently possible doing much more than this in order to encourage active participation. However, we should keep in mind that, in many situations, passive participation is a much more common behaviour than active one. It is perfectly possible that many people are happy just browsing MO or finding MO answers by googling their questions, and are not interested in having a personal interaction with the site (to make some analogies, there are many people that like to watch soccer in tv, but are not interested in playing themselves, or that are passionate about politics, but not actively engaged with any party). After all, in 10 years of MO we collected a huge amount of material, and the answer to many natural questions is already there and easy to find.
In fact, when the non-participation is independent on our behaviour, I find much more difficult to make effective proposals.