To make sure that a child's biological parents are also the legal parents, both mother and father should be listed on the child's birth certificate. If you want to add a parent's name to a birth certificate, contact your state's Bureau of Vital Statistics. You can find this contact information by visiting the website of the National Center for Health Statistics (click on the link "Where to Write for Vital Records," part of the "NCHS Top 10" list on the right).
In order to be listed on a child's birth certificate, most states require unmarried fathers to sign an affidavit or acknowledgment of paternity. In any case, it's a good idea for both parents to write, sign, and notarize a statement acknowledging the father's paternity. You can take this one step further by contacting your state's Vital Statistics office and asking whether they keep paternity statements on file. If they do, make sure yours is filed. The declaration operates as a judgment of paternity in most states.
Rules for same-sex couples can be different in some circumstances. For more, see Lesbian & Gay Adoption and Parenting.