Which (almost) balanced Boolean function has smallest "total" influence
You have the lower bound of $\mathbf{Inf}[f] \geq \operatorname{Var}[ f ]$ for $f\colon  \{-1,1\}^n\to\mathbb{R}$ (Poincaré Inequality), so that for an (almost) balanced $f\colon  \{-1,1\}^n\to\{-1,1\}$ one must have
$$
\mathbf{Inf}[f] \gtrsim 1 \tag{1}
$$
(say, $1-\epsilon$, depending on the near-unbalanced assumption). Further, for a dictator function, this is tight:
$$
\mathbf{Inf}[\chi_1] = \mathbf{Inf}_1[\chi_1] = 1\,. \tag{2}
$$