Is there a way to check if a path in a graph is infinite?
You will not find an algorithm to do what you want. As you indirectly suggest, an algorithm can only check a finite prefix of any path.  You can't even use the process of elimination as there may be multiple infinite paths. (If there was only one, you could eventually rule out all the other finite paths.)

As you are probably aware, this is (a corollary of) König's lemma.  What you are asking for is essentially a constructive proof of it which doesn't exist.  The typical (classical) proof repeatedly uses double negation elimination in the form: if no child has an infinite (simple) path then the subtree has a finite number of vertices, this is a contradiction thus there must be some vertex that has an infinite number of descendants. There is not, in general, a computable function to pick out this vertex though.  Knowing that it must "exist" doesn't mean we have any way of calculating it.