The tomato you picture has rotted spots all around the fruit, not just on the bottom. It doesn't seem to match anything except cold damage. It is possible there was damage from our cold evenings several weeks ago.n   The leaves could also be exhibiting signs of cold damage but it also could be an early blight. Lower leaves should be removed in any case. You should not have them touching the soil. Soil contains fungus which can be transferred when it splashes up onto the leaves. You can check available information at this site: http://www.extension.umn.edu/garden/diagnose/plant/vegetable/tomato/index.html

Thank you for your help. The angle of the picture makes it appear that the leaves are toughing the ground, but in reality they are not. However with the rain we have had, they still could have been splashed. I will remove the lower leaves, but I was wondering if all the affected fruit should be removed also.

Yes, remove the blemished fruit.  It will only get worse, not better.