I live and work in Arizona for a business based in New Mexico. Do I owe state taxes in both states?

If you only worked in Arizona, you should only file/pay tax in Arizona.  If the New Mexico taxes were withheld, you should file a nonresident NM return and report 0 income to get a refund.  If you have not done so already, contact your employer and request no NM withholding.

In the section "Taxes paid to another state or country" on the Arizona tax return, do I say nothing or enter the amount that was paid to New Mexico?

I suggest you begin with fresh data imported from the federal return. If you are working remotely, you will owe taxes to Arizona, where you live and work, not where the company is located. 

1. First, please check My Info in the Federal return and be sure the box is checked for Made money in another state (even though you did not)
2. Then go to the States section and click on the delete trashcan icon for both New Mexico and Arizona
3. Then, add New Mexico first, and be sure it is a Non-Resident state and complete it
4. It will show your federal income and ask you to allocate what is sourced in New Mexico, which is 0
5. Next, add Arizona and be sure it is a Resident State
6. It will show your federal income and ask you to allocate what is sourced in Arizona, all of it
    

This should prompt the allocation of income in both New Mexico and Arizon. Both returns will ask what part of the federal income is sourced in that state, and since you should owe 0 taxes in New Mexico after you complete the return, you can respond with 0 to Arizona when it asks about taxes paid to another state.