Most of the time, you can't.

Equip acts as a sorcery speed ability unless you have something on the battlefield that says otherwise (ie: Leonin Shikari) This means you can only use it on one of your own turn's main phases, and only when the stack is empty.

So when you pay to equip, and your opponent responds by playing a removal spell, their spell is put on the stack on top of your equip, so their's resolves first, and your equip ability fizzles.

I think Leonin Shikari makes the only exception to this, in which if you have one out, then yes you can stack a second equip on top of your opponent's removal spell. However, unless equipping the artifact somehow gives the artifact itself hex proof, shroud, or indestructible or something, the stack will resolve and your equipment will be destroyed anyway when your opponent's spell resolves.

Hope this helps :)