You might pick up an algicide that's safe for animals that may wander by and have a drink. You can get that at a fishpond store or supplier. There are lots of brands which should work. A very small amount (maybe a couple tablespoons) will do the trick but you may need to add more every month or so during the hottest/sunniest parts of summer.
Introducing fish and/or plants could help take care of the algae as well if you're not into using the algicide. I would not recommend chlorine (altho. it will kill the algae) because it will kill beneficial bacteria that also help keep the fountain/pond clean.
Another approach, which is considerably more expensive but 100% effective, is to put an ultraviolet algae filter in the water line from the pump to the top of the water feature. For example, here's a link to the one I use in my ~250-gallon fishpond. The light is fully enclosed, so there's no visible light that leaks out and watertight so it can be submerged in the water feature to help hide it.

The water comes in one side, gets irradiated in the cylinder and exits on the other side. This example might be kind of large for the fountain, I think, but they do make smaller and cheaper ones. Do some shopping for the best fit, etc.
A UV light like this is 100% going to get rid of the algae in a day or two and prevent it from coming back. The one I posted as an example runs about 10W so it isn't something that's going to cost a lot to run. You probably don't need to run it all the time, but it won't hurt to do so.