Is the language $K=\{u \in\{0,1\}^n\mid n \geq 0, \forall_{v \in\{0,1\}^n} (u+v) \in L \}$ regular?
It might be easier to consider the complement of your language:
$$
\overline{K} = \{ u \in \{0,1\}^* : \exists v \in \{0,1\}^{|u|} \text{ s.t. } u+v \in \overline{L} \}.
$$

You can now easily convert an NFA for $\overline{L}$ to one for $\overline{K}$ by replacing all edges labelled $1$ with edges labelled $0,1$ and all edges labelled $2$ with edges labelled $1$.

I'll let you figure out why this works.