Nowadays, it has become accepted that when a woman is in labor, her husband is with her in the delivery room, and remains there for the delivery. Very often, the wife wants her husband there for moral support and encouragement as she endures the excruciating pain of childbirth. Is this Halachically permissible, or should the husband stay outside the delivery room?
This question was addressed already by Rav Moshe Feinstein (Russia-New York, 1895-1986), in Iggerot Moshe (Y.D. 2:75), where he writes that there is no Halachic basis to forbid the husband from being present when his wife gives birth. However, the wife becomes a Nidda during labor, and so the husband must observe all the restrictions that apply when his wife is a Nidda, such as ensuring not to see normally concealed parts of the body, and not to touch her. If the wife needs somebody to hold her hand, another woman, such as a nurse or doula, should be brought to her for this purpose. (In theory, if the wife would be in danger if her husband did not hold her hand, then it would be permissible, but this is unlikely to happen.)
We might add that the Mishna lists three sins for which women are punished when giving birth, one of which is laxity in the laws of Nidda. Thus, at no time is it more crucial for the couple to properly observe the laws of Nidda than during labor and childbirth, and so if the husband chooses to stay with his wife in the delivery room, he must ensure to strictly comply with the Halachot of Nidda.
Some women prefer having their mother with them during childbirth while their husband recites Tehillim, and of course, this is perfectly acceptable.
It goes without saying that it is wholly inappropriate to take videos of childbirth.
Summary: It is permissible for a husband to stay with his wife in the delivery room during labor and childbirth, provided that he adheres to the prohibitions that apply when the wife is a Nidda, as a woman becomes a Nidda during labor.