Almost all modern PCBs are washed in water or isopropyl alcohol to remove flux residue, especially that nasty flux that's needed for the crappy lead free solder you Europeans made us all use (sorry, had to say it

). They are usually then air dried but are sometimes baked at ~75C to encourage the moisture to go away.
However, new PCBs usually use LPI soldermask. This holds up to moisture well. Older PCBs used other kinds of soldermask which can, in some cases, cause the mask to fall off due to the moisture getting under it. I've never had this happen to me, but it can happen. You're usually going to be fine, though.
Be sure to remove socketed chips since moisture will get stuck around those sockets, and let things dry at least overnight (better for a couple days) before using again.
Yes, you can actually usually use a dishwasher if you want, but it's kinda harsh (but very effective). No detergent, turn off any extra heat during wash/rinse. Heated dry at your option; I recommend air dry. Keep the boards on the top rack.
As for air from an air compressor containing excessive moisture, I've never had a problem. I clean computers all the time. Heck, I do it while they're on sometimes, though that may be living a little dangerously. I've never seen water come out of my blow gun, but the air isn't "dry", no. A little water just isn't a big enough deal to care. You can get air line dryers if you want.