I like triple chelated copper based algaecides for fountains (no fish or plants). If the fountain was very white on you didn't want an aging look I'd go with ammonium. Not as effective against as many kinds of algae, but still good. These are very long lasting so you don't need to add stuff every few days.
Most copper is considered portable water safe at the directed dose, and that will be a tiny amount for a fountain. Dosing higher doesn't work any better and can hurt birds who come for a drink.
A "green" solution would be peroxide, something like GreenClean algaecide. The peroxide at the drug store might would, but you might need a lot, like a few quarts. The down side with peroxide is the algae can come right back. Products like GreenClean are much more concentrated but pretty expensive for me.
Chlorine is great for an initial treatment if there's algae present. Not my fav long term because you have to keep adding it and I'm not a fan of the odor especially if the fountain is in an enclosed courtyard. It takes a pretty high dose of chlorine to kill algae. For example the normal chlorine level in pools can still allow algae growth, but mainly the macro algae kinds than grow on the sides.
Salt isn't my fav because it takes really high levels (lots of algae species love high salt content) and you get scale build up which looks bad imo.
None of these will hurt most pumps, especially fountain pumps. Some large pumps have steel cases and so salt wouldn't be good for them. I don't know how big a fountain we're talking about.
Mosquito dunks will control mosquitoes.