N.J. Stat. § 52:28-25
Secondly. That each state shall enjoy and exercise a concurrent jurisdiction within and upon the water, and not upon the dry land, between the shores of said river, but in such sort, nevertheless, that every ship and other vessel, while riding at anchor before any city or town in either state, where she hath last laded or unladed, or where it is intended she shall first thereafter either lade or unlade, shall be considered exclusively within the jurisdiction of such state; and every vessel fastened to or aground on the shore of either state, shall in like manner be considered exclusively within the jurisdiction of such state; but that all capital and other offenses, trespasses, or damages, committed on said river, the juridical investigation and determination thereof shall be exclusively vested in the state wherein the offender or person charged with such offense shall be first apprehended, arrested, or prosecuted.
N.J.S. § 52:28-25