The main problem is for securing the blockchain you need something hard to compute but easy for any node to verify you computed it correctly.
So really for folding etc it makes more sense for them simply to be given all the coins in the first block then use them to pay people to fold, or do it like devcoin where they get most of the mined coins and, again, they simply then use the coins to reward folding.
The folding itself is useless for securing blockchains.
Basically the cosest you ould get would be solidcoin type thing where some master node(s) decide if your blocks are okay, so really there is no point relating it to a blockchain at all.
They might as well just issue people folingcoins or whatever in their own central database since only they can tell if someone folded properly so since they are the central authority anyway the insanely expensive overhead of securing a blockchain is simply a total waste of money, waste of enery etc. Blockchains are crazy-expensive, andlook at bitcoin for example everyu single coin ever created is being paid to miners for securing the chain, plus miners also get transaction fees, so basically more than 100% of all the money is the cost of securing the blockchain! It costs more to seucre the blockchain than the total number of coins ever created!
However if you can do like devcoin, use merged mining; or, just even use devcoin IF your folding and its solutions are all free opn source data and programs, I guess you would then have some coins to reward miners with.
You could just as well though create yet another pump and dump coin and give 90% of the coins to the folding authority, and let them use them to pay people who actually insist on being paid (lets remember a lot of volunteers only do it because they are NOT paid so they could even lose volunteers if they paid...)
The important thing to note though is the whole folding stuff is useless for blockchain securing.
All it amounts to is having a central authority that says yes or no to your blocks, and they can do that regardless of folding or no folding. So entangling the two together is kind of pointless, they might as well just give out bitcoins or litecoins or whatevercoins to people instead of folding points, without mucking with the blockchain stuff at all. Or probably much cheaper, pay people with some currency that does not have the massive overhead of a blockchain assocoated with it thus is much much less expensive to run.