My old computer is too slow for anti-virus software. What should I do?

You didn't indicate which version of Windows you're running. Given the age and characteristics of the machine, I'm guessing it's also an old version, and is itself no longer getting regular security updates.

That can be a problem. While "upgrade to XP" is the common solution, that's not going to work here. 128 megabytes of RAM just won't cut it.

Which bring me to perhaps by far the most common recommendation for older PCs.

Don't run Windows.

Install a version of Linux. Ubuntu, perhaps, but there are also several distributions of Linux that are tailored for smaller machines.

If what you're attempting to do can be done with Linux and applications available for Linux, then that's a very lucrative approach. Not only will your machine run much faster than any current version of Windows could allow, but you'll have sidestepped the vast majority of malware issues. Because of Windows' huge market share, almost all malware targets vulnerabilities in Windows. If you're not running Windows, most malware just isn't interested in you.

If for some reason Linux isn't feasible and you need to stick with the version of Windows you're running, all isn't lost. We just need to take extra care.

Update - as best you can, make sure that the software on the machine is as up-to-date as it can be. There's no reason in exposing yourself to issues that could be easily avoided by having the latest versions.

Get behind a firewall - an exceptionally large number of vulnerabilities that perhaps can't or won't be fixed in your old machine can be completely blocked simply by placing that machine behind a firewall. In this case, I strongly suggest a hardware firewall, like a router, just to avoid installing more things on your older machine.

Secure the rest of your network. You should be doing this already, of course, but this becomes doubly important as you now have a machine you know you can't completely secure. If another machine on your network becomes infected, that infection could easily jump to this older machine. Even if you clean up the original infection, the damage will have been done.

Scan downloads - you're already doing this, and it's a good & important step. Just be sure that the malware software and database you're using are up-to-date.

Consider a cross-network scan - not a strong recommendation, but something to consider. Some anti-malware tools will allow you to scan a drive that's been shared on a network. For example, you might temporarily share the root of the "C" drive, and then from another PC on your network running anti-malware software, connect to that share and run the scan. This could be slow, and not all anti-malware software will do it.

Be smart. I know you are, but I need this for completeness. Everything you know about internet safety goes double on this older machine, particularly things like paying attention to what you do, surf and run on this machine - making sure to never run anything even remotely suspicious.

Ultimately, the life of older machines can be extended much longer than most people seem to think. I know I have several that are well past their prime when compared to existing technology. I even have one that's a true workhorse for data storage on my home network.

And it's not running Windows.