It's a good idea to clean and dry after running in wet weather, only for the fact of rusting. The Batt should be okay, but for the future you should try to weather proof it a bit(tape it up/plasti-dip). When I get my batt's ready for the weather, I spray plasti dip on all the seams where water could get in and cause some corrosion or any other issues.
Blow dry it or I like using compressed air cans to get it all out, then re-oil all you barrings if you can and you should have no issues with rust/corrosion or damage caused by weather. Just try your best not to get you tire foams wet(damages beyond repair most of the time)