Volatility sensitivity of a digital struck at spot

A contract pays $1$ at expiry if the stock finishes above today's price $S_0$. How does its Black-Scholes value change as volatility increases?

It falls. The value is $e^{-rT}N(d_2)$ with $d_2=\left(\tfrac{r}{\sigma}-\tfrac{\sigma}{2}\right)\sqrt T$, which decreases in $\sigma$ for $r\ge0$; as $\sigma\to\infty$ the value tends to $0$. Higher volatility drags the lognormal median below the forward, so finishing above $S_0$ becomes less likely.