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.
travel.stackexchange.com/users/12046/Spehro Pefhany
A substitute for silica is rice - that is more easily available.
Submerge your passport in rice with all the pages separated.
travel.stackexchange.com/users/17142/missanita
One very fast and reliable way it to simply leave your open passport next to a fan - put a fan onto a table and put the passport onto some raised object in front of the fan.
Once the pages closest to the fan dry out
(which is like in half an hour)
you can proceed to other pages.
Alternatively you can interleave the pages with something like pens or similar size objects so that they are separated and then the process will go much faster.
travel.stackexchange.com/users/703/sharptooth
I've used tinfoil to separate the pages and stood it on the radiator this worked a treat : )
travel.stackexchange.com/users/33178/adrian
I just now found out that if you wash your passport and live in west Texas, you can just let it sit on a table in the back yard at midday.
That moisture will disappear by the time you finish your second Shiner Bock.
travel.stackexchange.com/users/43892/Osvaldo Montoya
Put paper towels between each page and a large pot with heavy cans inside to flatten the covers.
travel.stackexchange.com/users/101665/martin
I'm sure I'm not the first traveller to experience this, so here goes...  Hostel last night I had my passport in my laptop case with my laptop.  Unfortunately this was under the aircon unit, which leaked overnight, creating a nice pool of water, soaking said bag.  Laptop is wet but works.  Passport is not soaking but definitely has water seepage throughout.  What's the safest way to dry the passport without damaging? I don't want to risk making it unusable.
How to dry out a wet passport?
travel.stackexchange.com
<passports><security>
travel.stackexchange.com/questions/31392
travel.stackexchange.com/users/101/Mark Mayo
While passports are less vulnerable to water damage than before, you could dry the passport with paper towels between each page and hang it up to dry. You can also use silica gel or rice to soak up some moisture. In addition, you can dry it with a fan. Finally, you could separate the pages with tin foil and dry over a radiator. Paper towels and rice are highly recommended for drying out a wet passport. Silica gel is another suggestion but more difficult to acquire. Other reliable methods are a fan, a radiator and the midday sun in Texas.
Paper towels can be used to dry out a wet passport. Put a wet passport in a container with silica gel. Submerge a wet passport in a container of rice. The midday sun in Texas, a fan and a radiator are alternative drying methods.