MDF (particle board) edge sealing for smooth painted finish
You could try brushing the edges with glue.  I use Weldbond glue for similar purposes.  You can dilute it, maybe a thin coat first to penetrate and seal, followed by a thicker coat to finish it.  It's available in large sizes if you'll use a lot.  Worst case it's a good glue so even if it doesn't work for your application, it will still be good to have around.  



There is a craft product made for decoupage called Modpodge that might work too.  They have different formulations, one might work for you.  



Wood turners use cyanoacrylate (like SuperGlue) to harden and strengthen wood, usually a thin one that wicks well.  It imaging it may be expensive to use, though, I could see MDF soaking up a lot of it.  E-ZBond makes it in larger sizes that may be more practical.