For student parents at Berkeley, there are a number of services and programs available.

Housing:

There are 2 sites of 'student family housing' (which means both undergraduates and graduates, along with post-docs and some faculty). The larger site is in the city of Albany, just north of Berkeley, and is named "UC Village". The smaller site is "Smythe-Fernwald" and is located in the hills south-east of campus in Berkeley. You can find more information here:
http://grad.berkeley.edu/financial/student_family.shtml

As a student parent (once an undergrad with infants) and a long-time resident of UC Village, I highly recommend student-parent housing. Your neighbors will have collective experience with all things parenting and UC. Plus, you will have a supportive place to raise your children. They will make lots of friends and you can easily arrange things like babysitting co-ops or baby study groups. Spouses and domestic partners also find a lot of support in resident housing. There are many programs run by the community center in the Village, such as American 101, Mommy Tea & Me, Single Parents Group, and so on. There are also Village festivals, recreation classes with discounts for Village residents and more.

Childcare:

Student, post-doc, and faculty parents can also register for childcare through UC Childcare (Early Childhood Education Program). They offer facilities that care for children from infant-school age. Over the program's history they have offered various packages of financial aid for childcare services. See more information here:
http://grad.berkeley.edu/financial/student_family.shtml

Healthcare:

is now available to the dependents of students. See more information here:
http://uhs.berkeley.edu/students/insurance/families.shtml

Now, for the less obvious information:

If you are pregnant:

There are policies that support new parenthood (official leave policies for academic appointments + extension of 'normative time') and breastfeeding (there are a number of lactation rooms on campus). See this flyer for more information:
http://grad.berkeley.edu/publications/pdf/parents_guide.pdf

School Age Children:

If you have School age children, you will be able to register your child in the school district in which you live. These district boundaries are typically coincident with the city boundaries: El Cerrito (West Contra Costa County), Albany, Berkeley, and Oakland. Each school district has their own criteria and method for placing children into schools. You should investigate your own school district to understand better how that system works. For example, Berkeley uses a 'busing' program to mix the population within the city across the schools. Albany tends to place children based on preference and proximity.

After School Programs: many schools offer on-site after-school care. You can also find private or city-run after-school programs that will pick up your child at dismissal. Once you know what district you will be in and which school your child will attend, you can find out more about after-school care from the school, the district, the city recreation program, UC Childcare, or from parent networks like "berkeley parents network".