After our harvest last fall, I decided that I wanted to try and save some of our potatoes. I placed them in boxes in a dark corner of our basement in hopes of replanting in the spring. These came from cerfified disease free seed potatoes. They have been sprouting for a while now and I would like to know if I can use them this spring?
How should I proceed? Should I let them keep sprouting or remove the sprouts now or when planting?
Here is a picture of what they look like.

Don't know the "correct" answer but we've planted many with sprouts that long or longer.  We simply cut the sprouts to about an inch long and covered with a couple inches of dirt in the trench just like normal.  No problem with them surviving to maturity.  As long as there is no visible mold and the tuber size is about 2 oz or larger it should be no problem.

Thanks for the input, do you still cut them up into multiple pieces or just plant them whole?

It depends whether we cut or not.  We always try to minimize labor where we can.

We work backwards to determine whether to cut or not.  We use a 100' row length which would be 100 seeds at 12" spacing.  If we have 50 seeds available, we'll have to cut enough to make 100 seeds.  Obviously if we have 90 seeds available we have to cut much less to fill the 10 spaces vs 50 spaces.  So basically figure how much area you want to fill and go from there.

As mentioned before when/if you have to cut, don't make them smaller than about 2oz and leave a couple eyes.  If you ever have to buy seed figure about 1 lb per 5 feet of row.  

If you had/have a disease problem such as a blight on your tomatoes or potatoes, trash the seed potatoes and start fresh!

If I were you I'd plant them whole or only cut them in halves. I'd plant them ASAP. You can plant them in early to mid march and you will have good potatoes come july.

If you can get some dirt worked, plant them now. It wont hurt a thing.

If you go through them ever couple of weeks and rub the eyes/sprouts off of them they potatoes won't get so dehydrated.  Those sprouts are pulling the moisture out of the potato.

Then, when it comes time to plant, you can cut them up into pieces with one or two little sprouts on each piece and put them under the dirt in a well tilled area.  

My dad used to keep the biggest of everything as "seed" for the next year.  He set aside the biggest potatoes, the biggest ears of corn, and with beans, he keep the ones with the most beans in a pod.  When he started (over 30 years ago) he was saving beans from pods of 5 or more.  The last year he had a garden he didn't save any beans for seed unless they were from a pod of 8 beans or more.  This was for both green beans and soup beans.

I've planted from "last years" crop and it worked, but I buy new seed potatoes to prevent disease.  We may not always have a choice in the matter.

On the subject of disease be sure to rotate your potato location and do not plant tomatoes in that spot either the following year (or two years if you're following three year rotation like I do).

Good advice!