  In this paper, we give a systematic study of Seiberg-Witten theory on closed
oriented manifold $M$ with codimension-$3$ oriented Riemannian foliation $F$.
Under a certain topological condition, we construct the basic Seiberg-Witten
invariant and the monopole Floer homologies
$\bar{HM}(M,F,\mfs;\Gamma),~\hat{HM}(M,F,\mfs;\Gamma),
  ~\widecheck{HM}(M,F,\mfs;\Gamma)$, for each transverse \spinc structure
$\mfs$, where $\Gamma$ is a complete local system. We will show that these
homologies are independent of the bundle-like metric and generic perturbation.
The major difference between the basic monopole Floer homologies and the ones
on manifolds is the necessity to use the Novikov ring on basic monopole Floer
homologies.
