When the procedure is performed ex-utero, there are few direct dangers from the collection of cord blood from the umbilical cord and placenta following a birth, as the procedure is performed on what would have otherwise been medical waste at this point.
Cord blood collections performed while the placenta is still in-utero carry a small risk of injury to to mother (not because the procedure itself is considered dangerous, but because some risk is always present during any internal medical procedure.)
The primary dangers that do exist during cord blood collections come from two concerns:
After birth, while the umbilical cord is still attached to the child, blood still continue to flow to the baby. Clamping the cord immediately or too soon after birth can deprive the baby of needed blood supply.
The cord is normally clamped and cut regardless of whether the cord blood is being collected - however, in an attempt to harvest as much volume as possible the concern is it's clamped too early.
If clamped prematurely and the baby deprived of sufficient blood flow, there is an increased risk of anemia (cord blood contains significant iron), there is an increased likelihood of respiratory distress syndrome, and a number of other possible complications have been suggested (though not all have been validated through study).
The amount of time the cord should remain attached and unclamped is a matter of contention among practitioners - many doctors prefer to clamp the cord as early as possible to reduce the possibility of jaundice (regardless of whether there is cord blood collection), while there is a growing movement towards allowing as much blood to flow to the baby as possible before clamping, especially amongst natural birth advocates.
Because the cord blood collection typically occurs shortly after delivery, there are concerns that both the newborn and the delivering mother may not receive adequate postpartum care immediately following birth, as the attending physician and/or birthing practitioner will be occupied with the collection itself rather than the patients.
This is a legitimate concern of course, but need not be a problem assuming there are an adequate number of practitioners present, and that their primary concern and focus remains the mother and child.
While the above are certainly legitimate concerns that should be noted and discussed with your birthing staff in advance - cord blood collection is generally considered a very safe practice by most practitioners.