The resin will not harm your plumbing......if enough went down the pipe to either slow or clog the drain ,treat it as you would any other slow or clogged drain. Once the resin has cured and dried, the only way to break it down is to let it soak in acetone or rubbing alcohol and then peel it off of the sinks surface.
Brian's right, if it went down the pipes then it would only "coat" the pipes. Which, depending on how old and how narrow your pipes are, might not be bad as long as it didn't cause a clog - some pipes are fairly wide so it takes a very big clog to stop them.
I work in my kitchen (as it's the only room I can use right now) and when I spill any resin on the sink it's usually just little drips so I let it harden then use a plastic scraper to pop them right off.