Corroded PCB from Rechargeable NiMH battery
Is this exposed copper something I should worry about?


Maybe, in the presence of humidity the residual salts/acids that are on the copper and solder junctions could further corrode. The best way to corrode metal is with salts and water.  


  Is there some means I can use to protect the copper?


Yeah, go over everything with a soldering iron and protect all that copper with a layer of solder. Thats how regular PCB's are protected in the factory if you get a HASL surface finish (63% Tin 37% Lead) which is close to most solder compositions. 

As an added benefit, if flux is used in the soldering operation, it will probably help clean some of the bad residues out of the corroded traces. Flux will also prevent solder bridges while soldering. 

Don't burn the connector while soldering, either carefully unsolder, or in some cases the shroud can be 'slid' off while soldering and the installed after (be careful with that operation also if that's the route you go)