It's more likely to be regular root rot caused by drowning rather than pythium.
I don't know what the spots are, but they probably also have something to do with overwatering.
I think the best thing you can do is make sure the pot is draining well and nurture the plant back to health by making sure it has the right amount of light and water. Bright, indirect light (as from an East facing window) and moist-but-not-wet soil would be perfect.
I would also recommend that you don't give the plant any fertiliser for a while. If you do want to give fertiliser, use it at 1/4 strength.

People will probably write in telling you to repot your plant. This may be a good idea, but if you do end up doing that, just be careful not to damage the healthy roots the plant has left. Palm trees really don't like their roots to be disturbed.

Thanks PatchyJack,

I have applied these treatments so far however it seems like the situation keep deteriorating, with new fronds dying every weeks.

I have searched further about the various kind of root rot disease and it seems like in almost every case, you have to repot the plant in clean soil and remove / destroy every affected plants :(

I recently repotted the plant (when the soil previously had problems to dry out) Also, most of the palms seems to be affected so I don't really feel good about destroying all of them :(

On another note, it seems like some of the stems only have a brown patch on their outer layer so it might be possible that they will recover if the disease is only spreading from there.

Im thinking about attempting to remove every layers that are affected, it seems like the stems / frond that are linked to them are dying so I don't thinks it would be a really bad thing for the plant.

Edit : I have removed on badly damaged stem which had a rotting spot and it look like color is almost black near the roots. There was also some tiny white spots on them so it really doesn't look good. I don't really have any fungicide but I don't even know at this point if I should try to apply something such as peroxide or just remove as many of the affected plants as possible. This is definitively not looking good :(

This post was edited by ShaZe on Wed, Jun 18, 14 at 16:12

Areca palms are very easy to kill with overwatering and lack of light and humidity. They are not the easiest palms to grow as houseplants. They want more light than most. They are also prone to spider mites.

there is no reason to apply peroxide. Stop watering and wait and see if it recovers. Fussing with it isn't going to help. Sorry.

I agree. Just leave the plants alone for now.
If they really do have bad root rot like the one you removed, then there's not really anything you can do.
They may recover on their own, or they may not.
Just look after them as best you can in the meantime.

The other thing you should do is not blame yourself if the plants don't make it. It's sad when these things happen, but sometimes things go wrong with houseplants. It is the way of the world.
Best of luck!