Assuming the fit is not too tight, riding in the rain shouldn't be a problem. If the fit is tight and you were to allow the plug to be soaked, it could expand and potentially do some damage. Unlikely, but it could happen.

I would actually have issue with totally plugging the bottom of a steerer tube with something like a cork. That's a bit like a story told on here once about a sponge being found in a seat tube that had rusted completely through. You're blocking off water from entering going up. You're also blocking water from exiting in it's natural, gravity encouraged direction of down.