You could dry the pages with paper towels as much as possible, then hang it to dry on a string with the pages spread out or separated by bits of paper towel- don't leave them in contact with each other wet or they may stick together. Change the paper towels for dry ones periodically as it dries.
You could also put it in a container with some silica gel, but buying that tends to be a bit of expedition even if you live in the place you're looking (some craft stores sell it for drying flowers). Or put it in sealed container with (uncooked, of course) rice, but always keep wet pages from spending too much time in contact by using paper towels.
I dried out a passport and entire wallet contents this way with little trouble (the cell phone didn't fare as well, unfortunately).
Be sure to inspect any unused visas to make sure they've not been damaged.
I've found that passports (Canadian ones anyway) are amazingly resilient-- despite being soaked with water or sweat they're still usable with just a bit of wrinkling on the cover to show for it.