Your basic left click attack with a staff is considered an Attack like any other weapon, so getting +x damage on Attacks on jewelry will add that damage type to your staff attack as well. As far as I know, there are no staves that have anything but raw physical damage on attack, since they're focused on buffing your spells rather than your basic attack, which is not counted as a spell for some reason.

The above info is correct but I'll add in my opinion on staffs currently; they're kind of garbage. All other weapons in game buff their respective skills and melee and ranged attacks are based off of the actual weapon dmg. The weapon dmg on staffs only buffs your basic attack.
Not only that, since 2H weapons can only have 3 sockets (and 4 affixes max [for rares]) you get more gems + magical properties with dual wielding or using something silly like a gun + catalyst to cast spells.

You can buff your basic attack dmg as the poster above said by adding in gems or getting gear that says "adds x-y dmg to attacks" but if you're using a staff you're casting spells only...so why would you waste a precious gem socket or line on a gear piece.

The only reason I can think of is if you're doing some kind of multi-ailment build and having issues deciding what combo of dmg types you want on what spells or having issues getting the numbers right so one dmg types doesn't overwrite another in terms of ailment priority. You can add attack dmg gem to your staff to make it do a specific ailment.
It'll also change your basic attack to look like the dmg type like in this video.
https://www.youtube.com/watch?v=IQCrCsIWKvY
Although basic attacks don't exactly look like this anymore but it'll change the colour/effect.

My mage character has an Aether Dmg Staff. It still also does Material dmg but the majority of it is Aether. It also shoots purple bolts which is super cool.

On a side note. You currently cannot beat the last boss of act III if you use any Aether damage and most ailments. So take that into account when playing right now. :(

I like the effect when you attack with sword or when I shoot with bow. when it has toxic damage so the slash is green. Staffs haves only white effect attacks:( Otherwise, thank you for the explanation. I hope they add other effects on staffs

This is false, I played through the Story with an Aether/Shadow-focused Time Weaver/Abyss Shaper mage which inflicts tons of Stasis/Curse and killed each phase of the act 3 final boss in seconds with no problem. It only glitches out if you let him re-enter his apocalyptic form in phase 2 after knocking him out of it once; beat the phase before he tries to re-enter and no glitch. I've also heard that skipping cutscenes is what causes this bug so watching every cutscene leading up to the fight in full will prevent it, but no idea if that's true or not.

I don't know man. I am burn damage ailment build and I obliterated all the chapter bosses in seconds.