What other ways can the career of a Postdoc be made more appealing given a certain level of discontent at the moment? I don't have time for a silly and serious solution this time around so I'll leave it to you to decide where this one belongs.
If a postdoc can only realistically be a postdoc for 2-3 terms and if only 1 in 5 get tenure track positions (a figure for America in 2012) then a postdoc should really be honing skills for another career. There're two ways they can go about this - be super organised and do so in their "spare" time (impressive) or prioritise the new job over the actual postdoc position (bad). Obviously there could be a third option which would be to provide postdocs with training and opportunities to move into alternative science jobs while doing their postdoc. There are a lot of science related jobs out there from the obvious industry to publishing, patenting, teaching, science communication/media, tech development, government advisor/policy maker etc, etc, etc (there are a lot of science related jobs out there).
Step one should be to provide PhD students/postdocs with seminars/advice on all of these careers. Then the university/institute should actually help them with getting the relevant experience/tuition to maximise a smooth transition into said career. It shouldn't be too difficult as academia has strong ties to many/all of these types of careers and I imagine most of these careers wouldn't mind a way of tapping into potential talent. I'm sure a large hurdle for any postdoc at an interview for another career is trying to convince them this isn't your plan B and that you'd rather be in a lab still. Being able to point on your CV to how you've been moving towards this job for sometime benefits both parties.
I'm not suggesting it's the institute's priority to get the student/postdoc a job in another career but I do think it should be a responsibility. If 80% of your workforce's career stops cold the least you could do is help them move on (it also looks better on institute stats). The hard work ultimately has to be on the part of the postdoc but they need to be able to do so without there being any stigma that they are quitting/losers for considering an alternative. Ideally everyone should be made to spend time investigating an alternative career (even all those people who know they will become PIs because they really don't) that way it can't be seen as a stigma to spend time investigating and developing your skills for said career. For those who still get to be PI's it's not like it would be wasted time as most, if not all, of these careers can be of use to the PI in future. Understanding the publishing process from the other side or being able to recognise a lucrative patent/application of your work is no bad thing.
There seems to be a range of how seriously academia takes this. Most places take PI training very seriously with lots of seminars and workshops on all the things you need to get and run your own lab. There's usually quite a drop off for anything not PI-related though. Often they let you go off to a one-day careers fair for free which while an excellent first step should ideally have some follow up. My current university has a yearly series of "alternate science career" talks with opportunities to network with the speakers from that profession It also has a lot of one-off workshops that cover a variety of things in an A-team style where if you can find them you can go. They don't however insist you have to go to these and while you could argue that's fair there is the problem that students/postdocs don't attend any because they feel they don't have the time or shouldn't be attending such things.
The "stigma" is a general thing and is quite weird. Most PIs I know never vocalise such a sentiment (and I'm pretty sure don't harbour such thoughts) but wherever I've been there seems to be that feeling about alternate careers in PhD and Postdoc circles. It's an indirect thing in the sense people openly talk about applying for a new postdoc position, writing a grant or applying for a PI position but are much more secretive about looking for other jobs. I think this "stigma" could be alleviated if the places of research took alternative careers seriously.
Some institutes do take a very active stance on alternate careers. I know that the Netherlands Cancer Institute (NKI) has had such a program for several years (I assume it's still running) where they actively encourage employees to consider alternatives to being a group leader . I've also heard that the new Crick Institute in London wants to adopt a similar approach with active training/opportunities for people to explore alternative careers.
While critics would suggest it's moving the focus away from research I think it's a vital move for institutes that want to keep attracting students and postdocs. I'd argue that someone who wants to keep their options open is far more sensible than someone who recklessly assumes they'll make it. It could also be valuable in showing that career prospects for science are great without it being an optimistic black hole of people eventually finding that other science job by themselves. I'd also argue that the time lost from the bench is probably worth it in that people will be less worried about their futures with some kind of framework for an active backup and a relaxed mind is a more productive one.
If anyone reading wants to praise their place of work for some innovative approaches to alternate careers please mention it in the comments. They deserve the credit and free advertising :)