Not for an SO but if my husband was laid off I'd help. You'd end up helping indirectly anyway because if his unemployment does not cover the child support obligation, bills still have to be paid and money is still going to come out of your pocket to offset the cost of him losing his job. I know how the Court system works very well and would rather not spend money hiring an attorney to defend against a contempt motion...or worse...if you own a home or have joint accouns...they could garnish bank accounts, put a lien on your home (lis pendens), throw your husband in jail, revoke his license, garnish tax return refunds, etc. (although you have to be pretty severely in arrears for them to do all of this). You just have to weigh the amount of exposure you have to the consequences and make a wise decision based on your own self-interests.
If I did make any payments towards the support because we were married and it was in my own best interest, I would certainly be doing my own wage withholding once he found another job.
If his losing a job was his own fault, I would not help.