Q. Is it permitted or even recommended for a husband to be present when his child is born to provide support and encouragement to his wife during a difficult labor?
A. Minchas Yitzchok (8: 30) recommends that the husband should not be present unless it is a situation of safety or danger, in which his presence can be helpful.
Igrois Moshe (Y.D. 2: 75) argues that he does not perceive any prohibition if he does not contemplate the actual exiting of the baby, as he stands next to his wife's head.
Horav S. Z. Auerbach zt'l as quoted in Nishmas Avrohom, permits a husband to be present in the delivery room during birth as he does not violate any Halachos by doing so. However, he does caution that it isn’t a 'nice practice' (“Ain HaDavar Yafeh”) and should only be permitted if the wife is genuinely frightened or in some danger. A husband’s presence is objectively calming for a woman during labor and delivery, and is not merely a reflection of the demands of the modern age and popular culture.
Horav Aharon Miller's Shlit'a opinion is that there should be a curtain or similar partition between them.
Rabbi A. Bartfeld as revised by Horav Dovid Pam, Horav Aharon Miller and Horav Chanoch Ehrentreu Shlit'a