So the question is whether there is a sentence $\phi$ over the empty language such that for any structure $M$, $M\models\phi$ iff $M$ is infinite.
I’m pretty sure the answer is no, but I don’t see why. Compactness seems relevant, but I don’t see how.
HINT: consider its negation. This is a sentence satisfied in precisely the finite structures.
Now, there is a standard corollary of compactness: if $T$ has models of arbitrarily large finite size, what can you conclude?
We can also avoid compactness here: via Ehrenfeucht-Fraisse games, you can show that every satisfiable sentence in the empty theory has a finite model, and indeed we can effectively find a bound on how large this model has to be. For example, if $\varphi$ has the form $\exists x_1,..., x_n[quantifierfree]$, ...