  We present ionized gas kinematics for 681 galaxies at $z\sim 1.4-3.8$ from
the MOSFIRE Deep Evolution Field survey, measured using models which account
for random galaxy-slit misalignments together with structural parameters
derived from CANDELS Hubble Space Telescope (HST) imaging. Kinematics and sizes
are used to derive dynamical masses. Baryonic masses are estimated from stellar
masses and inferred gas masses from dust-corrected star formation rates (SFRs)
and the Kennicutt-Schmidt relation. We measure resolved rotation for 105
galaxies. For the remaining 576 galaxies we use models based on HST imaging
structural parameters together with integrated velocity dispersions and
baryonic masses to statistically constrain the median ratio of intrinsic
ordered to disordered motion, $V/\sigma_{V,0}$. We find that $V/\sigma_{V,0}$
increases with increasing stellar mass and decreasing specific SFR (sSFR).
These trends may reflect marginal disk stability, where systems with higher gas
fractions have thicker disks. For galaxies with detected rotation we assess
trends between their kinematics and mass, sSFR, and baryon surface density
($\Sigma_{\mathrm{bar},e}$). Intrinsic dispersion correlates most with
$\Sigma_{\mathrm{bar},e}$ and velocity correlates most with mass. By comparing
dynamical and baryonic masses, we find that galaxies at $z\sim 1.4-3.8$ are
baryon dominated within their effective radii ($R_E$), with Mdyn/Mbaryon
increasing over time. The inferred baryon fractions within $R_E$,
$f_{\mathrm{bar}}$, decrease over time, even at fixed mass, size, or surface
density. At fixed redshift, $f_{\mathrm{bar}}$ does not appear to vary with
stellar mass but increases with decreasing $R_E$ and increasing
$\Sigma_{\mathrm{bar},e}$. For galaxies at $z\geq2$, the median inferred baryon
fractions generally exceed 100%. We discuss possible explanations and future
avenues to resolve this tension.
