That is a very interesting question. In New York, judges have the discretion to order a child support
amount which is different than the child support guidelines
. The general feeling of the courts is that a non-custodial parent
is aware of his obligation to his child prior to his re-marrying or having additional children; therefore, the court will not typically reduce the amount of child support based on a new relationship or child. In addition, should the child support paying parent's income be reduced, the court has the ability to factor in the new couple's standard of living which basically includes the new spouse's contribution to the family, while technically not including the new spouse's income in the child support calculation.
In addition, some of the methodsd which can be used for enforcement of the child support order include garnishing the non-custodial parent's pay (I have even heard of judges threatening
to garnish a spouse's income - but have never seen it actually done and don't believe it can be done); attach unemployment benefits; interception of state/federal tax returns (even if joint with a new spouse); seize any lottery winnings; attach and seize financial assets (bank accounts); suspend driver's and professional licenses and passports; and file liens against any real property owned. Be aware that child support delinquency is also filed with the credit agencies and may very well negatively affect the abiltiy to borrow, purchase a home, etc.
Should you decide to marry this person, I would suggest that you continue to file taxes separately and maintain any of your separate property separately. Of course, if she/he is current on the child support and intends to keep current this will not be a problem.
Should she/he have a child support order that is too high, you may want to suggest that she/he contact the courts and attempt to have the amount reduced.