The OP lists email as a specific example, so I will provide an example with something other than email.
More consumers use text messaging and voice calls to communicate; at least I do when outside of work. Therefore, an app to protect those communications would cover most of what I send. There are a few apps that are cross platform and have a good pedigree to facilitate this.
I have had some success convincing some people to switch to these. I even offer to pay for the switch, but no one has taken me up on that. Most people are not staunchly opposed to it, but just don't want to be bothered. I have substantially reduced my non-secure traffic to non-secure users.
Something to keep in mind about securing cell phone traffic is the remarkable frequency in which users will switch phones. Upgrades, damages, etc. Those encryption/authentication keys will be lost and the users will not care.
Signal is an effort to make secure voice and sms-like messages easy to use and free to the user. It replaces the stock messaging app, and provides additional encrypted VoIP calls to other Signal/RedPhone users. Since this is free, looks like the stock app, and requires to complicated setup, this is a relatively easy pitch to other consumers. Signal is the merger of RedPhone and TextSecure.
Threema was available cross platform before Signal/TextSecure. I think it was available before Signal/TextSecure was on Android. This is an encrypted messaging platform and has been endorsed by Steve Gibbs of GRC in episode 409 of his Podcast Security now. However, it costs $2.49 and has strong authentication, which is rather complicated to setup: you scan barcodes on each others phones. On Android it's relatively easy, but on iOS it's always a challenge.
Getting to your auxiliary questions:
- Will people care?
- Should I try and enlighten people about the risks?
Do you know of a good website explaining in a simple language the
risks of unencrypted (mail) communication? I would place a link to it
into my mail signature.
- No, not generally. Only a few will be adamantly opposed though.
- To a certain degree. Broach the topic and let it simmer in their brains and offer to help get them started.
- See below.
I find more success using a meme, or referring to a 3rd party for the explanation. This way I have less skin in the game and they get the illusion of a more knowledgeable entity while getting it broken down barney style for them.
The EFF - have a page dedicated to this sort of thing, and there are even "Why should I care?" sections. This would be a good page to link to on an email signature:
EFF Surveillance Self Defense
Going back to email, there have been developments since this question was asked. I'm here because I was going to ask a similar question, so I'm fleshing it out to be THE Q&A.
There are now several options for encrypted email, specifically with GMail. You can even encrypt it in the webmail session!
I was unable to find anything describing the new feature that GMail can natively encrypt your emails without an extension. I wonder if I am remembering some news wrong. I had thought there was a new feature in GMail that was a 1 click button, built into the GMail web GUI, that changed your message to an encrypted one.
The easiest way to start gaining traction is for the initial setup, maintenance, and any migrations, to be easy on a nontechnical user. As a technical user, I have put off PGP because I know it's an involved process. Apparently it's gotten a lot easier, so I may revisit this for my own communications.