The media itself shouldn't matter too much. The most common for human cells is RPMI 1640. It's the differentiation factors (cytokines) that you add that makes the difference. Hopefully, someone will be able to give you more detailed information.
As for the type of serum... it's up to you. Usually for in vitro work, and FBS is still your best bet. Human serum is WAY too expensive for you to use just to figure out culturing conditions.
I never worked with stem cells, but if you go with what worked for bone marrow cells, at least for monocytes/macrophages it's GM-CSF or M-CSF.
Check our paper in Blood Bedi et al 2009 113:317-327 for culture conditions for eosinophil differentiation in vitro from hematopoietic progenitors.. Should also work for basophils, but would check papers by the Judah A. Denburg lab for basophil differentiation.