Many parents of infants place a monitor near the baby’s crib and a speaker in another room so they will hear if the baby wakes up. The question was posed to Hacham Ovadia Yosef as to whether parents may set up the monitor before Shabbat, such that on Shabbat they will hear the baby’s cries in the speaker in their room when the baby wakes up.
Hacham Ovadia penned a lengthy essay on the topic, in his work Halichot Olam, addressing the more general issue of the status of microphones on Shabbat. He explains that it is clearly forbidden to use a microphone on Shabbat, however, the prohibition involved is only "Mi’de’rabbanan" – a violation of a Rabbinic enactment. When a person speaks into a microphone, he certainly does not do anything that could be considered "kindling" or "extinguishing" in any halachic sense. At most, he increases the flow of electricity, which would be forbidden only "Mi’de’rabbanan."
Addressing the specific case of a baby monitor, Hacham Ovadia notes that the fact that the baby’s voice activates the monitor’s system is certainly of no concern, because the infant obviously does not intend to activate anything. And as for the fact that the speaker loudly sounds the baby’s voice in the house, Hacham Ovadia notes that this poses no problem according to Sephardic practice. The Shulhan Aruch allows setting up the millstones before Shabbat even though they will make noise on Shabbat, because in his view, there is no prohibition against having noise in the home on Shabbat.
One could argue, however, that using the monitor should be forbidden because the parents might talk to the baby or to each other in the baby’s room, and they would then be activating the monitor’s system through their speech. Hacham Ovadia dismisses this argument, however, noting that this situation falls under the category of "Pesik Resheh De’lo Ichpat Leh" – an action which inadvertently results in a Melacha which one has no interest in whatsoever. Although the parents’ speech would inevitably increase the flow of electricity in the monitor, this is neither their intention nor their desire – and therefore, since the prohibition at stake is "Mi’de’rabbanan," and not a Torah violation, speaking is allowed.
As such, Hacham Ovadia permits the use of a baby monitor that was set up before Shabbat, and even permits the parents to speak in the baby’s room over Shabbat. He notes, however, that some monitors are sold with an apparatus that covers the monitor so one can speak when it is on without their voice being detected, and it would be preferable to use monitors with such a system. But even without this apparatus, the use of a monitor on Shabbat is permissible.
Incidentally, Hacham Ovadia in this context also discusses the status of a hearing-aid on Shabbat, and concludes that one may wear a hearing-aid on Shabbat, despite the fact that a hearing-aid essentially works as a microphone, amplifying sound. Hacham Ovadia writes that he consulted on this matter with technology experts, as well as with Rav Shlomo Zalman Auerbach (Jerusalem, 1910-1995), who was widely considered among the leading experts in the field of electricity in Halacha. He concluded that since speaking to the hearing aid at most increases the electric current, it does not entail any Torah violation, and thus may be permitted, as discussed above in regard to a baby monitor.
Summary: It is permissible to set up a baby monitor before Shabbat to hear the baby in a different room, and it is even permissible to speak on Shabbat in the room where the monitor is placed.