Yes, you can use α-MEM with phenol red but without HEPES to grow mesenchymal stem cells with slight modifications. Add sodium sulphtae and KCL (0.005%) for oH adjustment.
Minimum Essential Medium (MEM) α is widely used for mammalian cell culture as well as selection for transfected DHFR negative cells. MEM α can be used with a variety of suspension and adherent mammalian cells, including keratinocytes, primary rat astrocytes, and human melanoma cells. Life Technologies offers a variety of Gibco® MEM α modifications for a range of cell culture applications.
The type of media used depend on cell type and may vary from cell to cell type (since you didn't describe the MSC type so requirement can't be defined in a cell specific way) but yes, MEM is the most widely accepted media for stem cell applications and proper cultivation and you have no need to use HEPES, sodium sulphtae, KCL, Glutamax, high glucose OR ANY OTHER ADDITIVE. Infact to be cost effective you have no need to go for even readymade MEM (it's good if your budget allows that, in that case you just need to buy a good quality FBS). You can use powdered media (MEM) from a good company (I use Sigma) and supplement it with normal antibiotics and sod. bicarbonate and use as such but take care that it should contain some non essential amino acids as that support the growth of stem cells very well.
One point I want to focus on is quality of FBS. It doesn't matter which media you use or that is less important than a good quality FBS so you can combine MEM with US origin FBS to obtain high proliferating cells. I have tried and standardised the same after years of experience as I have done primary culture of the same..And in my point of view these are the best conditions to grow MSCs.