Personally speaking, I'd open the case very carefully, writing down the exact steps I take to take it apart so I can follow it in reverse to put it back together.
Then I would carefully and gently clean the entire circuitboard and battery compartment with cotton swabs dipped in rubbing alcohol.
Then I would take the front & back panels and wash & dry them.
I would let it sit and air for a bit to make sure there was no moisture on any of the parts.
Then I would put it all back together again.
Then to ensure nothing like this ever happens again, I'd seal it into a clear plastic ziplock bag and use the remote through the bag, replacing the bag as it wears out.
Something to think about here:
You don't just have plain water inside it...you most likely have soup in there as well. As that dries, there will be a residue of grease, salt, and other particles left behind unless it is opened up and cleaned. And depending on the type of soup, it could also be acidic. This isn't good for the electronics inside.
Taking a hair dryer to it without opening it is likely to just create steam inside which will condense as it cools and put you right back to where you are now with it being wet inside. Not only that, but the heat could damage it, causing the remote to never work again. It also leaves all the crap from the soup in there to eat away at the circuitboard and possibly corroding it and your battery compartment.