According to the Wikipedia article "Conscription in the United States", The Selective Service Act of 1917:
...prohibited all forms of bounties, substitutions, or purchase of exemptions.
This national conscription process was not under state control, and effectively replaced the state-based quota system which had been in place during the Civil War and earlier, and which had proved generally ineffective.
After this date, the New Hampshire constitutional provisions would have applied only to conscription for service in the New Hampshire State Guard (NHSG), a purely state-organized force which was not authorized for service outside the boundaries of the state. Although still legally authorized, the NHSG has not existed as an actual force since 1947, when its last armory was turned over to the National Guard. Proposals to reactivate it moere recently have not passed the legislature.