You can allow supporters to opt in to cover their tickets' processing fees. Do this by selecting Processing fee comes out of the charge in the first column, which will make the checkbox for Give registrants the option to pay the processing fee during registration appear. 

Checking the box will make a new field appear, in which you can write your own custom prompt:

When supporters reach the checkout page, your custom prompt will appear as a preselected checkbox:

If they uncheck the box, they will not pay the processing fee; your nonprofit will absorb it.