Here's what I understand from Basis.
Basis: it's set of linearly independent vectors which can span the vector space.
Basis for Zero vector space:
case 1: when { 0 } , it's singleton set , since there's none in it to compare with to check linear independecy hence it can be considered as basis and also it can span it.
( i.e 0 = c . 0 , where c is any constant )
So, the basis has 1 vector, so dimension will be 1.
case 2: { any vector K } again it's singleton set , since there's none in it to compare with to check linear independecy hence it can be considered as basis and also it can span it.
( i.e 0 = 0. K (the vector K) )
Again, the basis has 1 vector, so dimension will be 1.