What is the safest way to deal with loads of incoming PDF files, some of which could potentially be malicious?

Safest would probably be a burner device. Grab a cheap laptop, and a mobile internet dongle, use it to download the documents, and manually copy across any contents to your main computer (literally retyping would be safest, if you're particularly worried). Since it's not on your network, it shouldn't be able to cause problems even if it got infected, and you'd be able to wipe it or just bin it if you have any particularly evil malware sent to you.

If you need actual contents from the files (e.g. embedded images), one option would be to install a PDF print driver on your burner device, and to print the incoming PDF files using it - this will generate PDF output, but, in theory, just the visual components. Printers don't tend to need script elements, hence they can be safely dropped. Bear in mind that some PDF printer drivers spot when you provide a PDF, and just pass it through unmodified - test before relying on it! Once you've got a clean PDF, email it back to yourself, and check with a virus scanner on your main machine before opening. Note that this doesn't completely eliminate the possibility of malware getting through, but should minimise the chances.

So, I try to stick with these concerns in the "land of reasonable". With every security issue there is a balance of secure v.s. safe. For example, you could buy a laptop, read one PDF loaded from the web mail side of your email provider, re type any content you need on a "main computer" then destroy the laptop starting all over again with a new laptop. That would be pretty secure. Also costly, and a giant pain.

So back to a "reasonable" approach.

First, use Linux and a up to date PDF reader. By doing so you have really reduced your exposure. There are not as many viruses written for Linux as there are for windows. That alone will protect you quite a bit. The viruses that do work on Linux are more complicated to implement. Again reducing your exposure.

Next use a Virtual machine that supports snap-shotting. The idea is that you setup your Linux OS inside a virtual machine host (like VirtualBox) get it all setup then, "Snapshot" the state.

You can then do all your "risky" work inside the virtual machine. Using isolation options, I don't know of any virus that can "escape" the virtual machine and get to the host machine (doesn't mean they're not out there, just means it's more rare, and more complicated for the attacker).

At the end of the day, or any time during the day when you think you have gotten a virus, then you "revert" the machine to the previous snapshot. All the changes and data that "happened" after your snapshot are undone, including any work, virii, etc.

During the day, you can open a PDF, scan it with ClamAV (or the like), copy and paste what you need, or what ever you need to do with the PDF files, so long as your Virtual Machine exists in isolation. That means that you don't give the virtual machine access to the host machine. You use something like email to transfer the files. Maybe FTP between the host and the virtual machine. Something, but not direct integration. Not dropbox either. Something where if you're going to transfer the file, then you're only going to transfer that one file after you're pretty sure it's safe. If you're using a Linux host and a Linux guest then scp is a great choice.

This gives you a "pretty secure", disposable environment, to check your questionable PDFs out, with the ability to "undo" damage that may happen, without having to really change much in your work flow.

Virtual machine hosts and guests can be almost any OS including Windows. Keep in mind that if you have a Linux guest and a Windows host the Linux virtual machine may not even be susceptible to a virus that is in the PDF that a Windows machine will be susceptible to. Scanning with an anti-virus scanner is important, no matter the OS combo in use.

Converting all the PDFs to some more "passive" format - maybe TIFF or postscript - could be done in batch, in a restricted account either on the local machine or on some linux box/VM. An exploit/malware being carried along into a different file format is very unlikely.

Files that are purely malicious will not even render that way; any exploit targeted at popular PDF viewers probably will not work with scripted conversion tools (which will mostly be based on the ghostscript engine); and the restricted account will keep a successful exploit from doing much damage.

A normal user account on an up-to-date linux machine is very difficult to "break out of" - do make sure that this machine doesn't have unregulated internet access though, since network access is the hardest to control.

If disclosure of the contents of the valid PDFs would have dire consequences, make sure only one PDF at a time is accessible to the account running the interpreter at a given time (eg by copying the file into a staging location from yet another user account, running the interpreter via su/sudo (not sudo to root!), then taking the result file away. Rinse, repeat.

Oh, and: Keep the original files away from any (especially Windows) PCs that are set up to do previews of files in Explorer, in email clients or similar frontends!

Depending on your threat model, even the "burner device" or virtual machine approach might not be sufficient. If an attacker is looking to identify your location, or even if a spammer wants to validate that your email address is active, then having the PDF phone home after being opened will expose you. Crafty PDFs might even contain worms to infect other machines, though I've never seen that in the wild.

Thus, after downloading the PDF, you may need to disconnect the device from the network before opening it.

Just get a Chromebook. Problem solved.

I believe this to be the most pleasant solution as you get a new, cheap and affordable toy that works better than your other computer.

Chrome OS considers every program to be 'hostile' so everything is totally sandboxed.

You can run Android apps on recent Chromebooks too, plus you get the PDF viewer built in so there is nothing to install.

You can cut and paste from your PDF files and into some Google docs thing to break stuff out to make it accessible on your 'proper computer'. (The one that can get virus and malware problems).

It is all about using the right tool for the job. Chromebooks are built on a modern OS and do not have to support legacy cruft from 1999 (as other operating systems do).

I would recommend a "isolated" device only for downloading and opening the pdfs. Ie. not connected to the rest of your network.

Then print it (paper can't transmit malware).

After that you could scan it, then you have a copy in image format. The printer should be isolated and connected only to the contaminated device aswell in this case. The scanner can be connected to the rest of your network.

If you want a faster workflow you could just transform them to images and then send them to yourself if you need to view them somewhere else, though you have to gurantee that the mail is not infected in some way. No links/images/javascript injected and the file format isn't executable or pdf.

Which means that the receiving end needs to view only text no html or javascript.

Open the files in a sandbox.

If you're infected, the infection should be contained and you can reset this virtual environment.

Edit:

A sandbox is a virtual environment which you can create and remove at will, like a virtual desktop where you can do anything and then reset it back to the initial stage.

Infected? Simply reset.

Avast antivirus (paid version) for example has a sandbox solution which allows you to right click on the file and open it in a sandbox.

There are tons of solutions like this, another example is Sandboxie

Edit 2:

Removed the MD5 suggestion, it's inferior to the sandbox solution.