Do I report 1099 income if I paid the person in 2016 however they cash the check in 2017?

You should report the payment as made in 2016 if the money left your control in 2016 (such as, you wrote a valid check, dated in 2016, and mailed it in 2016).  If you did not mail the check until 2017, or you post-dated the check, or the check was uncashable due to NSF until 2017, then it is a 2017 payment even if you wrote it in 2016.

Generally, if the recipient received the check in 2016 they must report it as 2016 income even if they did not cash the check until 2017.  If the recipient truly received the check in 2017, they can report it as 2017 income.  They may get a letter from the IRS about the missing 1099-MISC, which they would answer by showing the canceled check, postmarked envelope, proof of deposit, or other information that they received the money in 2017 and will report it in 2017.  Or they could report it as 2016 income.  But that's not your responsibility. Just issue your documents correctly from your point of view.

This kind of calendar straddle is not uncommon.