Mazal tov. Of course, the best idea is to daven (if possible, with a minyan) before starting to watch the baby. We understand that this does not always work out; this response deals with such a situation.
Firstly, it helps to know the needs and habits of the baby (which is difficult, as they often change as fast as you learn them). Many babies will wake, cry, and fall back to sleep by themselves. Others whimper relatively calmly for a few minutes until their parents come. In such cases, it is best to reach a place in tefilla where one can stop before going to the baby, if one can concentrate. One does not have to stop davening at the first cry, nor does a parent need to interrupt another activity immediately when doing so would cause difficulty. If the baby continues to cry bitterly, the rule that the needs of a child are like those of a sick person applies.[1]
The preferred course of action depends upon the particular place in the tefilla that one finds himself. During P’sukei D’Zimra and Kri’at Shema, one can take a break to walk, motion to (which cannot be done in the first section of Kr’iat Shema[2]) or make sounds to a child.[3] These are all forbidden during Shemoneh Esrei without an acute need.[4] It is a halachic problem during Shemoneh Esrei to hold a baby or any other item in a manner in which one may be concerned that it will fall. This problem may apply to various other parts of tefilla as well.[5]
The following are the priorities when you can’t wait. If you can go to console the baby or rock her back to sleep and then continue davening, this is preferable. (If you think it is likely the baby will stir and need to be put back to sleep before you can continue davening, stand near the baby for Shemoneh Esrei in order to avoid the need to walk to the crib. At other points in the davening, it pays to not be so close, as you don’t want to lose concentration at the baby’s every twist and turn.) It is best to finish dealing with the baby first, provided that, by doing so, the time it takes you to recite the entire section of the tefilla you are in will not have elapsed.[6] However, if need be, you can take longer than that and continue from where you left off,[7] while being careful not to talk. It is better to hint and make signs and noises than to move from one place to another during Shemoneh Esrei.[8]
If it is likely that a very long time will pass until the baby will allow you to daven properly, then you can hold the baby while finishing to daven if that will quiet her enough to enable you to concentrate. This is because one who davened while holding something that he is afraid may fall does b’di’eved (after the fact) fulfill the mitzva.[9] The reason this is not optimal is that one’s concentration is compromised. Thus, if the only way one can concentrate at all is by holding the baby, then that should be done. A better idea under these circumstances may be to put her in some type of baby carrier.[10] This is usually very soothing for the baby, especially when one sways. Logic dictates that, assuming there is no chance the baby will fall out, this would be the best option, since one’s concentration would not be adversely affected, and the restriction on holding a baby during tefilla should not apply.