There are quite a few things you can do to put people at ease. They're not necessarily after something to take with them - they're wanting to feel confident that they have dropped their PC off somewhere reputable and to know they'll get it back. For shops with a store front they get that feeling - they know where their PC is, they see lots of other PCs when they drop theirs off, they see large signage out the front that cost a lot of money etc. In other words, they know that shop isn't going anywhere. I used to work in a repair shop and in the three and a half years I was there I was never asked for a receipt for drop off of a computer.
Obviously you need to find other ways to give people that nice warm inner glow when they leave their PC with you. Others here haven't had the same experience as you despite running their business from home so there are things you need to adjust. This is a good chance to look at how you work as if people are wondering whether you will still be there tomorrow, they will also be questioning your technical ability.
A few suggestions:
* Look at what you wear. If it's jeans and a t-shirt then invest in a couple of polo's with your logo embroidered.
* Have a landline number (VoIP is cheap). People associate that with permanency.
* When someone drops their computer off to you put their details in your computer system while they are watching you. If you already have their details, double-check their phone number(s) before they leave. Also ask how they prefer you contact them (phone, email or SMS). Little things like this give people confidence - they know you're asking these questions because you will be contacting them. Once done, print off some paperwork with a job number on it and tape it to their PC. Again, make sure they see this so they know you are organised and keeping track of which PC belongs to who. Also, print a copy of the job number and give them a copy. Tell them they can call you at any time and quote that number and you'll be able to tell them what stage their repair is at. Also ask them how soon they are wanting their PC back. All this focus on how you will contact them, when you will contact them and how soon they can expect to hear from you is reiterating that you are professional, organised, and will get back to them.
* If people come to your home to drop off computers, consider getting a sign with your company name and logo.
* If you are picking a PC up from the clients site there are a few things you can do:
1) Don't seem too keen to take it. Even if you think it will need taking with you to reinstall Windows, spend 5 minutes to boot it up at the customers site.
2) Once you've taken that look you can then say "Yep, it's as I thought. The best way to proceed is going to be to back everything up, then to reinstall Windows, all of your software and to restore your files. It normally takes several hours to backup and restore your data, and a couple more hours to reinstall Windows. With my current workload I'd expect this to take about 3 days - is this ok or do you need it back sooner than that?". By being quite verbose about what will happen you have shown the customer that this is something you do all the time because you know exactly what the process is and how long it will take. You've also explained the reason why you need to take the computer with you (I doubt many customers will want you sitting at their house for three days!), and you've kept them in control of the conversation (rather than being forceful about taking the computer with you).
3) Give them a business card, with your landline and address on it. Also explain to them when you are available (which might be any time of the day if you work for yourself, but tell them you generally answer the phone anywhere up to about 7pm). As always, this is about setting expectations upfront. By talking about when they can call, you are reinforcing the idea that you are contactable and aren't going anywhere. Try to do this before the conversation gets to the point where you're talking about taking their PC with you.
* Look at the way you pick up new clients. For me it was almost entirely word of mouth so I had no question of integrity because a friend or relative had already vouched for me. If you don't have this, try to build your brand. Ask to leave business cards in local take away shops etc. Invest some money in Google AdWords (it doesn't have to cost a lot to get lots of impressions). Consider a couple of magnets with your logo to go on your car. The more time and money you've invested in these things, the less likely it is you will be seen as a potential fly-by-night.