If the water got inside the voice coil gap it will possibly short the speaker out and burn up the amp.
Test it with an ohm meter and press on the cone to see if the ohm measurement changes a little bit.
If it measures good then its may be fine.
But even if there is just dampness that suddenly collects into a droplet and makes a short during playtime... Your amp will shut down or burn up.
Not all speakers will be affected by this because allot of them coat the voice coil with a resin.
if the spider is damp or wet... Id seriously let it sit for a week and make sure its totally dry. The spider holds the speaker wires in place. The speaker wire can short from a damp spider.
Yes I shorted out one of my mids before because of the water coming through the window gasket. I ordered those XTC Foam baffels from Parts express bvut ended up cutting them in half basically to cover up the tops of the speakers.