This thread is linked with I found two usb sticks on the ground. Now what?. The other thread includes some non-technical considerations such as innaM's answer, which suggests that the contents are none of your business and you should simply turn it in for return to the owner, and Mike Chess's answer, which mentions that the drive could contain government secrets, terrorist documents, data used in identity theft, child pornography, etc., which could land you in trouble for having it in your possession.
Other answers on both threads address how to protect yourself from malware while exploring the contents, but those answers won't protect you from a "killer USB", a key point posed in this question. I won't rehash what's covered in other answers, but suffice it to say that all of the advice about protecting yourself from malware (including rubber ducks, which inject keystrokes), applies.
Value and Brand Name
But I would start with Christopher Hostage's point about flash drives being too cheap to be worth the bother and risk. If the drive is unclaimed by the owner, and after considering all of the warnings, you decide that you just have to try to make it safe and usable, start by considering the value of the drive. If it is a low capacity, standard speed, no name drive of unknown age, you could replace it with a new one for a few dollars. You don't know the remaining life on the drive. Even if you restore it to "fresh" condition, can you trust its reliability or remaining service life?
Which brings us to the case of an unclaimed drive that's officially yours, and:
- it is a high capacity, high speed, brand name drive of recognized reliability and performance,
- appears to be in new condition, perhaps a recently released product so you know it can't be very old.
One point of these criteria is that the drive could actually be worth more than a trivial amount. But my recommendation would be not to mess with anything else for a second reason. As Journeyman Geek points out in a comment, rubber ducks and USB killers come in common-looking packages. The brand name packaging is hard to counterfeit without expensive equipment, and tampering with a brand name package in an undetectable way is difficult. So limiting yourself to familiar, brand name drives offers a little protection in itself.
The first question is how can you physically connect it to your system safely if it could be a killer USB, and that's what I'll focus on.
- The first clue is the drive, itself. There are miniature styles that are basically the USB connector plus just enough plastic to have something to grab to get it in and out. That style is likely to be safe, especially if the plastic has the brand name on it.
- Flip style drives are popular for rubber ducks, so be particularly careful with them.
- If it is a standard size thumb drive large enough to hold killer hardware, inspect the case for signs that it is a counterfeit or has been tampered with. If it is the original, brand-labeled case, it will be difficult to tamper with it without leaving signs that would be visible with magnification.
The next step would be to isolate the drive from your system. Use a cheap USB hub that you are willing to sacrifice for the potential value of the thumb drive. Even better, daisy chain several hubs. The hub(s) will provide some degree of electrical isolation that might protect your very expensive computer from the "must have", free killer USB drive.
Warning: I have not tested this and have no way of knowing the degree of safety this would provide. But if you are going to risk your system, it might minimize damage to it.
As LPChip suggests in a comment on the question, the only "safe" way to test it is using a system you consider disposable. Even then, consider that almost any working computer has the potential to be useful. An ancient, under-powered computer can be loaded with a lightweight, memory-resident Linux distro and provide some amazing performance for routine tasks. Unless you are retrieving a computer from the trash for the purpose of testing the flash drive, weigh the value of a working computer against the value of the unknown drive.