Yes. It is legal and I have open carried in the airport a few times without getting hassled.
The only place carry is restricted by NH law is inside courthouses (see RSA 159:19 below). (Federal law restrictions still apply obviously.)
Also don't forget that you cannot open carry a loaded weapon in a vehicle unless you have a NH non-resident license or a CCW permit of a state recognized by NH (Alabama, Alaska, Arizona, Arkansas, Colorado, Florida, Georgia, Idaho, Indiana, Kentucky, Louisiana, Michigan, Missouri, Mississippi, North Carolina, North Dakota, Oklahoma, Pennsylvania, Tennessee, Utah, Wyoming). (see RSA 159:4 below).
I hope you enjoy your trip to NH.
Section 159:19 159:19 Courthouse Security. –
I. No person shall knowingly carry a loaded or unloaded pistol, revolver, or firearm or any other deadly weapon as defined in RSA 625:11, V, whether open or concealed or whether licensed or unlicensed, upon the person or within any of the person's possessions owned or within the person's control in a courtroom or area used by a court. Whoever violates the provisions of this paragraph shall be guilty of a class B felony.
II. Firearms may be secured at the entrance to a courthouse by courthouse security personnel.
III. For purposes of paragraph I, "area used by a court'' means:
(a) In a building dedicated exclusively to court use, the entire building exclusive of the area between the entrance and the courthouse security.
(b) In any other building which includes a court facility, courtrooms, jury assembly rooms, deliberation rooms, conference and interview rooms, the judge's chambers, other court staff facilities, holding facilities, and corridors, stairways, waiting areas, and elevators directly connecting these rooms and facilities.
IV. The provisions of this section shall not apply to marshals, sheriffs, deputy sheriffs, police or other duly appointed or elected law enforcement officers, bailiffs and court security officers, or persons with prior authorization of the court for the purpose of introducing weapons into evidence and as otherwise provided for in RSA 159:5.
V. It shall be an affirmative defense to any prosecution under paragraph I that there was no notice of the provisions of paragraph I posted in a conspicuous place at each public entrance to the court building.
Section 159:4 159:4 Carrying Without License. – No person shall carry a loaded pistol or revolver in any vehicle or concealed upon his person, except in his dwelling, house or place of business, without a valid license therefor as hereinafter provided. A loaded pistol or revolver shall include any pistol or revolver with a magazine, cylinder, chamber or clip in which there are loaded cartridges. Whoever violates the provisions of this section shall, for the first such offense, be guilty of a misdemeanor. For the second and for each subsequent violation of the provisions of this section, such person shall be guilty of a class B felony, provided such second or subsequent violation has occurred within 7 years of the previous conviction.