WTVQ reports (thanks to Andy Banducci for the pointer); the bill text is here. Pretty clearly unconstitutional, see McIntyre v. Ohio Elec. Comm'n (1995) and the cases on which it relies. Plus of course any such state law would likely violate the dormant Commerce Clause, because it would end up affecting speech throughout the whole nation (given that even national ISPs would have to implement such policies for all their users, because national ISPs "do business" in Kentucky).
I should note that the First Amendment doesn't categorically protect anonymity: If someone sues you, or the government tries to prosecute you, based on your speech, and there's a credible claim that your speech indeed falls into a First Amendment exception, the plaintiff or prosecutor can indeed get subpoenas aimed at uncovering your identity. Likewise, the Court has upheld certain kinds of identification requirements for expensive speech related to candidate campaigns, and for corporate speech even about ballot measure campaigns; and identification requirements of some sorts may indeed be constitutionally permissible. Nonetheless, the constitutional rule is still that anonymity is presumptively protected, and a blanket ban on online anonymity such as this one would be pretty clearly unconstitutional.
Finally, let me mention that this is another area in which generalizations such as "suppression of speech has become a liberal monopoly" are mistaken. The legislator who introduced the bill is a Republican, and the two Justices who would have generally rejected a right to anonymous speech in McIntyre were Justice Scalia and Chief Justice Rehnquist. And this was so even though the law applied to clearly political speech; even "suppression of [political] speech has become a liberal monopoly" (an argument tailored to exclude matters such as sexually themed speech) are inaccurate here.
Of course, in McIntyre, other conservatives and moderate conservatives — Justices O'Connor, Kennedy, and Thomas — voted to protect the right to anonymous speech. And where the spending of substantial sums of money for speech is involved, liberals tend to be more willing than conservatives to require some sorts of disclosure (though many conservatives support such disclosure as well). But that just helps show that both the "liberals want to suppress speech / conservatives want to protect speech" and the "liberals want to protect speech / conservatives want to suppress speech" arguments are generally not sound today, at least when put at that level of generality.