Finite automaton whose alphabet is $\mathbb{N}$
In the standard definition of a finite automaton (say a DFA or an NFA), the alphabet is a non-empty finite set (perhaps the empty set is allowed by some definitions). The reason we insist that the alphabet be finite is that otherwise the description of the automaton would potentially not be finite.
For example, in one common definition of DFAs, the transition function is a function from $Q \times \Sigma$ to $Q$, where $Q$ is the set of states. If $\Sigma$ is infinite, then describing the transition function would require us to list infinitely many values of the transition function.
Some definitions of DFA allow the transition function to be partial, and in that case, we could have an automaton with a finite description even when the alphabet is infinite; and the same always holds for NFAs. However, in such cases all words accepted by the automaton belong to $\Delta^*$ for some finite $\Delta$, and so the effective alphabet is finite.

While the usual definition requires $\Sigma$ to be finite, the definitions of DFA and NFA still make sense even for infinite alphabets. However, if you keep the set of states finite, then you can find a partition of the alphabet into finitely many subsets $\{\Sigma_\sigma : \sigma \in \Delta\}$ so that the language accepted by the automaton behaves as a language over $\Delta$, in the sense that whether a word is accepted or not depends only on which partitions $\Sigma_\sigma$ its letters belong to (that is, it is obtained from a language over $\Delta$ by applying the inverse of the homomorphism mapping all letters in $\Sigma_\sigma$ to $\sigma$).
Allowing infinitely many states doesn't really change the picture: if the set of states is allowed to have the same cardinality as the alphabet, then all languages are regular, and otherwise the &quot;effective alphabet&quot; has the same cardinality as the set of states (as in the case considered above), and all languages over that alphabet can be accepted.
This shows that even from a theoretical perspective, the most interesting case is that of finitely many states and (&quot;without loss of generality&quot;) finite alphabet.