Place eggs and white vinegar in large pot. Top with cold water; bring to a boil. Reduce heat and simmer for 12 minutes. Remove eggs; chill.
Peel and small dice the beet. Keep the nice cuts; puree remaining pieces with a little water. Place eggs in beet puree for 1 hour to add color. Remove eggs; halve.
Combine yolks, mayonnaise, mustard, paprika, and apple cider vinegar; whisk until smooth. Salt and pepper to taste. Place mixture in egg whites; garnish with diced beets and basil.