G.B.
I would start by just really washing it well. If that didn't get rid of the stuff then I would sand it and do what Tracy K said. The final coat of Polyurethane is the deal breaker though. If you don't do that it will pretty much be in vain. That finish will allow you to wipe it down and keep it clean without having to do this much work again.
BYW, all of my grand-kids went to toddler beds around 15 months. They didn't realize they could get up and get in to things so they just thought it was for sleeping. We didn't have any issues at all. I also just kept a baby gate in their doorway so if they did get up they did not have access to any other part of the house. I did keep door know protectors on the bathroom doors and those doors shut at all times though as an extra precaution. Kids can't get their big heads up out of the toilet if they fall in, their necks are not strong enough until they are older than 2 1/2 to 3.