An Artillerist adds their Intelligence modifier as a bonus to the total damage rolled per hit.
The relevant text in the Artillerist subclass says:
Any damage roll you make for a cantrip in the wand gains a bonus equal to your Intelligence modifier (minimum of +1).
The bonus is added when the Artillerist rolls any amount of dice to calculate damage. The bonus is added once to the total. "Damage rolls" are defined in the combat rules:
Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target. Magic Weapons, Special Abilities, and other factors can grant a bonus to damage.
With Fire bolt, and most other cantrips, the Artillerist would make one attack roll. If it hits, they make one damage roll, and add their Intelligence modifier. At 5th level, that roll would be 2d10 + (Intelligence modifier) fire damage.
Although Eldritch blast involves multiple damage rolls, it's not a valid candidate because the Wand Prototype feature only applies to artificer cantrips:
When you do so, you invest it with one artificer cantrip of your choice—even one you don’t know—that has a casting time of 1 action.
However, if you had some way of making Eldritch blast an artificer cantrip, then the Artillerist would add their Intelligence modifier to each attack's damage roll.