Children are the purpose of life for millions if not billions of people. They are wonderful. A PhD degree is also a wonderful privilege, available to less than 1% of the population (at least in the US). However, a combination of these two experiences is a very significant challenge, both for the family (your relationship with your wife and child) and for the PhD study, or the chances of having a satisfying, engaging, and overall positive experience in the program.
I do not have kids yet but will share what I do know. I am aware of several peers in grad school who had to manage graduate study and a small child. Even with an older (say, school-age) child it is a difficult task. But with a baby the time commitment, unexpected as well as everyday expenses (diapers, food, clothes, medical care) present substantial added 'overhead' for a family. Perhaps the greatest demand a baby will impose is the demand for constant attention and time, which must be provided NOW, and cannot be postponed for a time and day that works best with your class schedule.
I don't know if you have a working spouse whose income can sustain a baby (including babysitting), or if you are thinking of studying and sustaining a family on a graduate assistantship (formally a 20hr/wk commitment, which in reality is often a 30-60hr/wk commitment, if done right). Add to that a 24/7 commitment to a baby...
Bottom line, yes, it probably can be done. But the odds are going to be against you. It will require immense time management, discipline, flexibility, lots of sleepless nights (both for the baby and for writing papers for classes), constant stress, and non-existent time for yourself and thus declining physical and mental health.
How is this different from having a baby with a regular job? A couple things.
First, the time commitment is different. Many jobs allow a fairly predictable 8-5 schedule. When you walk out of the building, you are DONE for the day. Not so with grad school. When you get home from the university, homework awaits! Budget 4-6 hours daily just on catching up on reading and writing for class, and either grading and preparing for lectures (if you are a TA) or doing research-related analysis/writing (if you are an RA). During my first 3-4 years of PhD study I rarely recall a less than 12-hr workday.
Second, finances. A full-time job that pays at least $50K a year, plus additional income from the spouse, can allow to raise a baby. It will not be a smooth process and will require sacrifices and trade-offs, but at least it is feasible, and the money can be stretched to cover some basic expenses for a baby (with frequent trips to the Dollar Store, Goodwill, and Salvation Army stores).
If you end up in a large university in a large city with high cost of living, you might make around $1400-2000/month for a 20hr/wk assistantship. The exact amount often depends on discipline (engineering/medical vs. business vs. social sciences vs. humanities). This is just around the official subsistence level (i.e. poverty line) income for a household in the U.S.
If you do something relatively unrelated to your program (e.g. an assistantship in a Writing Center or Career Center, etc.) then count on standard university assistantship. In a high cost of living large city, count on around $1500/mo. The figure is much lower if you end up in the middle-of-nowhere, small college town - think not much over $1000/mo. Usually, graduate students struggle throughout their time in school. I mean, carefully budget food and clothes, not to even dream of living in decent, clean, apartments with no roommates or having cars (insurance? gas?) or even considering pricey entertainment or travel. Those who afford these things usually have savings or rely on some other financial assistance ("free money" like scholarships/fellowships) or regular financial support from the parents.
Unless your wife plans to work and bring in substantial income (in the U.S., I would say at least $50-60K/yr), it would be extremely difficult, if not impossible and destructive to family financial situation, to try to raise a baby on a single graduate stipend. I strongly advise against even considering this as an option. A small medical emergency that is not covered by graduate insurance (which is often seriously lacking for dependent care) can put you in debt for years if not decades. This is before mentioning all the human cost of living below subsistence level. If you think the loving, caring, gentle relationship with a spouse will be unaffected by persistent financial hardship, this perception is incorrect.
I already mentioned the high demand on time and attention associated with raising a child. Well, graduate school by itself is like having a child. Your thesis, if you will 'survive' through the preliminary exams and enter the dissertation stage, will be just like another child that needs constant nurture and attention. It will be on your mind day and night, it will cause some stress and it will cause late nights. Many late nights.
Bottom line. Babies are wonderful. Doctoral study is wonderful. But in combination, they will require trade-offs and impose a substantially higher level of stress, and reduced quality of life, than if attended to separately. Since a baby is guaranteed for you at this point, my personal advice would be to focus on raising your child at least until school age (5-6), and revisiting the idea of PhD at that point. You will have a much more pleasant life as a parent with a baby, rather than a baby PLUS a thesis, to come home to every day. The PhD will be there and you can do it later in life, no problem. So, enjoy your baby, it's a wonderful gift. The rest is trivial in comparison. If an academic career is in your future, you will find a way to pursue it sooner or later. AT the end of your life, you will not wish you had spent more time in the lab or submitting grant proposals or conference papers. You will wish you had spent more time with your family. And I hope you do. Good luck!
