Yes you might want to put a Lacquer or Polyurethane to protect the wood from water stains.
If it is a gloss black paint and not stain, then you may not have to do anything else depending on the type of paint.
It is generally a good idea to seal up any wood that is painted on one side, not just an aquarium. If you don't, there is a chance the unfinished side will absorb moisture from the air causing it to expand/bow.