In a matched sample of Seyfert and quiescent galaxies we simultaneously probe the stellar and molecular gas kinematics from 1 kpc down to 50 pc with the aim of identifying the dynamical processes dictating black hole accretion rates. This first paper compares the global characteristics of a sample of ten galaxies. We find several differences within a radius of 500 pc that are correlated with Active Galactic Nucleus (AGN) activity. The Seyferts have: (1) a more centrally concentrated nuclear stellar surface brightness with lower stellar luminosities beyond a radius of 100 pc, (2) a lower stellar velocity dispersion within a radius of 200 pc, (3) elevated H2 luminosity out to a radius of at least 250 pc, and (4) more centrally concentrated H2 surface brightness profiles. These observed differences can be interpreted as evidence for Seyfert galaxies having a dynamically cold (in comparison to the bulge) nuclear structure composed of a significant gas reservoir and a relatively young stellar population. This structure is undetected (and possibly does not exist) in quiescent galaxies. The presence of such a nuclear structure in Seyfert galaxies provides evidence for inflow of the surrounding interstellar medium since the nuclear stellar population requires a supply of gas from which to form. The fueling of a Seyfert AGN is thus associated with the formation of a dynamically cold component of gas and stars on scales of hundreds of parsecs. ; Comment: 18 pages, 22 figures. Accepted for publication in the Astrophysical Journal. Minor revisions to match published version