Upgrade your book to water-resistant Rite-in-the-Rain® paper on the inside. It's exactly what you think – regular paper soaks up water droplets, this paper will resist them.
Oddly enough, Sharpie-type pens might be the best for this paper, with one caveat: the same witchcraft that makes the paper water-resistant also keeps ink from absorbing quickly, which can lead to smudging.
Baseball moves slow enough, it likely won't be an issue for most users. Once the ink is dry, it's locked in. But fast writers (and left-handed scorekeepers) may want to think twice on this paper choice.
As always, test your pen(s) on the cover sheet to see what will work the best.
** I would love to offer this paper with the Small Ball book, but those books require precise production – just not easily doable with the equipment I have. **