Is there a good way to find out, and possibly drain this water out?
A drill with a concrete bit should do the trick.
Will a dehumidifier work in this situation?
A basement in most climates (any place with somewhat humid summers) should likely have a dehumidifier regardless. So it definitely won't hurt.
This could be a DIY project, but it's pretty big. Anytime you start messing with support columns you can get yourself into a big project. But, that said, this is most likely doable.
You need to get the bottom of the post off the ground with some sort of concrete footing and replace the bottom (or all) of the post.
Basically, if you hired a contractor, he'd prop up the roof using doubled 2x8's on each side of the post. Then he'd either replace the whole post or cut off the post about 2 feet from the ground. He'd probably (based on code) also dig a hole and pour a concrete footing and then bolt the new post base back in. The size and depth of the footing can vary by local regs and the contractor's preference.
If it were me, I'd prop up the porch, take out a post, dig a hole (2 foot+, unless you are concerned about frost heave in which case 4 feet+), put in a sono tube up to about a foot above ground, fill it with concrete and put a new post in.
Based on local code, this would probably require a permit (but depends on where you are and the nerve of your contractor - personally, I'd do this myself without a permit...)
You reuse the metal bracket by embedding in the top of the new footing (sand it and paint it with rust proof paint), but you might not need it. Home Depot has lots of various bracket to attach vertical columns to footings. Also, google images is your friend on how this can look.
Best Answer
If built properly, a porch floor slopes away from the house, and (being a porch) is also covered by a roof, so it should only get incidental blowing rain, not the full brunt of every storm. The slope (1/8" per foot, 1 inch in 8 feet) ensures that any water that lands on the porch drains off of it.
It's also common to oil (linseed oil) or paint the boards as well.