This question is pertaining to the Janani Suraksha Yojana initiative in India, which is a safe motherhood intervention under the National Health Mission driven by the state governments. The accurate assertion out of the given four options are:

1) It is indeed an intervention by the State Health Departments aimed at ensuring the well-being of the mother during childbirth.

2) Its primary objective is to reduce maternal and neonatal mortality, specifically targeting poor women to help them gain access to proper medical facilities and treatments during pregnancy and childbirth.

3) The scheme promotes institutional delivery, meaning the birth is assisted in a health facility such as a hospital or health center, rather than at home where risks are higher.

4) However, the fourth statement is incorrect. The scheme does not directly provide public health facilities to sick infants up to one year of age. Although there may be indirect benefits, it`s not a core objective of the initiative.

Therefore, only three statements are correct, making option 3 the right answer.