Power Function IR Remote always emitting after putting batteries
The Lego Company is usually very good about replacing broken and defective parts for free.  In the case of expensive electronic parts such as this, in my experience, they will sometimes request you send them the broken/defective one.  If this is the case, they will make it easy for you and send everything you need to do so with your replacement.  Because this part is still made/sold, there is a good chance they will make it right.  

If this solution doesn't work out for you then you could try fixing it.  I would recommend taking it apart, if you have the tools, and look for any obvious signs of contaminants, over-heating, breakage, etc.  If there aren't any obvious issues then try storing the electronics in a bag of rice for a couple days to absorb any moisture in them.  If none of these things work then you are probably going to need a new one.