If your family has a history of genetic disorders, or cancers where a blood transplant could help then it might be a good idea to have your cord blood stored. When you bank it privately the blood will be:
1) immediately available (sooner the process the better)
2) Contamination free (since the disease most likely wont manifest till later in life)
3) A genetic match (which is important b/c it is not always easy to find one).
These points are all critical when it comes to needing a blood transplant and are not always east to find.
This was not an issue for us so we decided to bank publicly. We didn't want the cord blood going to waste when it could help save someone's life.
Three things to remember when going this route:
1) Ask your hospital if they will collect it for free. They most likely will. If not though you have to arrange the donation with a public bank (this will be done free of charge)
2) It is not an ethical issue. The embryonic stem cells that hear debated about in the news are completely different from the stem cells found in the cord blood.
3) finally, you will not be able to have access to the cord blood after it is donated. The process is completely anonymous.
You also might want to check this site out, it was a great source of unbiased information: http://www.publiccordbloodbanks.org/
Best of luck with your decision, and congrats!