I found these answers when I Googled.

Marine paint usually has all sorts of toxic chemicals in it, to prevent algae, barnacles and worms. I doubt it's good for a birdbath/fountain.

It looks like a latex water-based paint is the best choice.

Q: Is It Safe to Spray Paint My Bird Bath?

A: Yes, you can spray paint your birdbath. But do not use any spray paints that are oil or petroleum based. Make sure the bird bath is completely dry before painting. You should apply at least 2 coats, allowing each application to dry before applying another. Krylon-brand spray paints, among others, are non-toxic when dry. Use a clear coat spray paint as the final coat to act as a sealer.

And this

When I made my birdbath (using clay pots), I used terra cotta craft paint/glaze from the craft store. It's specifically for use on clay pots and concrete. Non-toxic and safe for birds. But, you've to seal it with a water-based, non-toxic sealer. Otherwise, it could flake off.

The paints and the sealer can be purchased at any craft store. Since they're more into carrying products for crafting projects than a home improvement, hardware or paint store....they're more apt to carry these paints specifically for such a project like the birdbath painting.

Also, when you clean the bath, make sure you don't scrub too hard. Even though it's "sealed", it could damage the painted finish.