My thought is try to stay close to original. Paint will raise questions if/when you decide to sell.
It's hard to see the damage in your picture, but unless there is rot or delamination, I'd at least have a try at restoring the finish. First correct the water problem. Wash, then wipe with alcohol. Dry the wood with warm heat lamps if required. Apply stain if needed then teak oil. You can get a sense of what the finish is going to look like if you wet the surface with mineral spirits (before oiling or varnishing). You will be amazed at what can be accomplished this way.
Bobk