Maybe this is too late to reply since this was a few months ago but I just saw the question.
There is a system of daycares throughout Baselstadt that coordinate with the state kindergartens to provide all day care for working parents.
www.baslerfrauenverein.ch
There may be a long waiting list. When we registered I was given priority because I'm a single mom and had to work but I know other families with both parents working that found a place. It helps to get your name on the waiting list early.
I think they open at 6:30 in the morning and close at 6 pm. The staff take the kids to kindergarten and pick them up for lunch and after school. When my daughter was a small I often joined them there for lunch too.
My daughter was in a daycare center (Tagesheim) in Klein Basel from 3 months and we were very happy with it. The staff were very well trained and caring. It was really more like a pre-school.
My daughter's kindergarten was assigned based on the nearness to her daycare center. She switched to an all day primary school at age 8 so she didn't need the daycare anymore, but some friends of hers still go for their lunch and after school homework help still at age 12.
Kate