If the mother wants to relinquish the baby to the state or to a private agency, they must give the father notice and a chance to object. The state can only terminate the father's rights against his will if he is found by a court to be an unfit parent, or that he has shown a substantial lack of regard for his parenting obligations and is withholding consent to the adoption contrary to the best interests of the child. (RCW 26.33.120)
If the mother relinquishes the child within 72 hours of its birth at a designated safe haven, the state still has an obligation to try to locate the father, but there is no guarantee that it will be successful if the mother does not want to cooperate, and the adoption process can move forward without the father after reasonable attempts have been made to notify him. If the father is concerned about this happening, he should contact DSHS and make himself available for genetic testing if he believes an unclaimed baby may be his.
Every case is unique; you should consult with an attorney privately about your particular situation. Opinions offered should not be relied upon as legal advice.