Yes, Hoechst can stain live cells. Hoechst 33528 and Hoechst 33342 are both cell membrane permeant, which makes both suitable for staining live cells. Both dyes are also less toxic than most other dyes, which ensures a higher viability of stained live cells.
Of the two Hoechst dyes, Hoechst 33342 is the preferred choice for staining live cells. This is because its additional ethyl group renders it more lipophilic, which allows it to cross intact cell membranes more easily. Within the cell, the dye binds to AT-rich regions of DNA in the minor groove. The fluorescence increases significantly on binding to DNA.