My Mum is nearly 82 years old and has no interest in doing it herself, although I think if she wanted to she would be able to. I buy a lot of things for her online, at her request, the last thing being tickets to fly to NZ to visit her sister. She has also looked at things we've shown her online and been interested, especially when I can tell her how much the bungalow up the road sold for, more or less than the asking price! I think if my Aunt in NZ had a computer, then Skype would have been enough to convince her to learn. She has no need, as I deal with all of her utility bills, so she just lets me do it. This is to do with her setting up direct debits - OK - but then leaving them for years, so she is then paying a lot more than she needs to. We would have set my Aunt up with a computer in New Zealand, but as she lives in Christchurch, she often doesn't have a working phone line, so a bit pointless for the foreseeable future. As she can't sell her house, she can't move back to the UK, which is her long term aim now she is a widow. They were both amazed when I bought and arranged delivery of a multi-region DVD player for her, from a local NZ department store, via the internet (pre-earthquake, shop is no longer there). I can then send DVDs via Amazon for her presents.
I have helped older people get online via my work as a support worker. My top tip would be don't try to help anyone if you do not have lots of patience, as you will have to constantly have to show or tell them the same thing many times until they get it. You also need to be both fairly competent and confident yourself.I write down basics for them, such as how to log on, how to open and save a document, how to access their emails, how to log off and close down properly. I've even taken photos and printed them with arrows drawn to certain buttons, or something on the screen, with a written explanation if it is a sticking point with them.
I use an automatic password and form filler - I personally use roboform. It saves lots of hassle especially when different websites have different criteria for passwords. Explain the importance of not using the same password for things like paypal as you do for other sites (banks have complicated criteria so you are unlikely to use the same information there as another website).
DO NOT click on any popups or adverts EVER. Also watch links other people post, especially in forums, as they can be malicious or just sick.
Use an online email account such as hotmail or yahoo, so that you retain access if you change internet providers or whilst you are away.
I also remind them that it becomes easier the more that they use it, they should not be frightened of either the computer or the internet, the latter they cannot break and the former can almost always be fixed, unless it's been dropped or it has had liquid spilt on it. And yes, I have witnessed a laptop being used as a tray for drinks 
If they have an interest such as steam trains, animals, genealogy, keeping in touch with distant relatives, then show them how they can pursue this online. Doing something that interests them is a really good way to engage their interest, it will motivate them to learn.
If they don't feel confident in backing up the computer find someone who will do it for them, also checking the security is up to date and unnecessary items are deleted. I know you can set them to run automatically, but I've used lots of different programmes over the past 17 years, and have yet to find one that doesn't need some manual checking up on.
All local further education colleges have beginners computer courses, the ones for complete novices are usually free or very cheap and are well worth it for those who are able to attend.
I only work part-time, but I am known as the 'tech support', any problems get left until I'm back in work (or I get a phone call at home, but I don't mind). I also have to sort out the malfunctioning freeview and Sky boxes. It is usually very simple to sort out, but they are too 'frightened' to have a go. I always try to show them what they need to do and some of them have actually done what I've shown them, if it has happened again. The biggest problem is when Windows crashes, I can usually get it working again - I haven't been defeated yet! I've lost count the number of times I get told that someone has broken their computer, when it is 'just' Windows crashed again (usually after an update).
After all I was a beginner once and I try to remember what helped me.