  Globular clusters are dense aggregates of stars that evolve in relative
isolation. For the better part of $40$ years these clusters have been known to
possess unique chemical signatures called abundance anomalies. Recent
observations have found these abundance anomalies to be the result of distinct
stellar populations with the youngest population undergoing an unknown
enrichment process. Understanding these chemical signatures requires a precise
understanding of the thermonuclear reaction rates at relatively low
temperatures. At these low temperatures reaction rates suffer from large
uncertainties arising from poorly understood resonances in several key
reactions. Transfer reactions provide key constrains on nuclear inputs for
these resonances. This thesis presents an updated understanding of the sodium
and potassium destroying reactions $^{23}$Na$(p, \gamma)$ and $^{39}$K$(p,
\gamma)$, respectively. A reevaluated rate for $^{39}$K$(p, \gamma)$ indicates
that it is less precisely known than previously thought, and future
experimental study is needed to reduce its impact on globular cluster
nucleosynthesis. Novel Bayesian techniques assess the uncertainties arising
from transfer measurements. These techniques are applied to the transfer
reaction $^{23}$Na$(^3\text{He}, d)$, which was carried out at Triangle
Universities Nuclear Laboratory. Results of this experiment indicate that the
energy of an important resonance in $^{23}$Na$(p, \gamma)$ is much lower than
previously thought at $E_r =132(3)$ keV. The transfer measurement also
indicates tension between previous direct studies of the resonance strength and
the current transfer measurement. The impact of these uncertainties on the
$^{23}$Na$(p, \gamma)$ reaction rate is investigated, and it is shown that this
rate requires more intensive study to provide the precision needed to constrain
nucleosynthesis in globular clusters.
