The same threat was utilized to get most states to pass seat belt laws.
So that's why New Hampshire doesn't get highway money.
In the case of the seat belt law (as opposed to the drinking age), the federal policy was that a certain percentage of states had to pass seat belt laws or there would be stronger federal standards regarding air bags put into place, or some such thing. I don't think New Hampshire was ultimately denied federal highway funding, though it probably doesn't get much in any case because it has little traffic flowing through it.