I think it's very nice of you to step up and pay it, however why isn't your wife doing it? She needs to be the bigger person and step up to her responsibility. If she is not sending the check herself I can only assume she is not visiting her child. The courts only care if payments for child support stop, not who sends check.
No, if ur spouse doesnt work because you decided that they wouldnt, then I dont think its wrong to pay their child support. If you make more money than them and are helping them out, while ur spouse does other bills. As a parent you know a child still needs things in life, whether its paid by you, the parent the child does stay with or the state. If you dont mind taking this responsibilty, then kudos to you and be your spouse's partner.