We study the initial-boundary value problem for a Keller-Segel-Navier-Stokes model that includes a logistic source term. The system is considered in a two-dimensional bounded domain with a smooth boundary as follows $$\begin{aligned}\,\, \left\{ \begin{aligned}&n_{t}+u\cdot \nabla n=\Delta n-\nabla \cdot (n\nabla c)+\mu n-\kappa n^{2},\\&c_{t}+u\cdot \nabla c=\Delta c -nc,\\&v_{t}+u\cdot \nabla v=\Delta v -\gamma v+n,\\&u_{t}+(u\cdot \nabla ) u+\nabla \pi =\Delta u-nf,\\&\nabla \cdot u=0. \end{aligned} \right. \end{aligned}$$ This system characterizes the interaction of chemotactic microorganisms with an incompressible fluid. Under the conditions of positive $\mu$ and non-negative $\kappa$, sufficiently regular initial data $(n_{0}, c_{0}, v_{0}, u_{0})$ with $n_{0} \not \equiv 0$ yield a uniformly bounded global classical solution. Moreover, when $\mu =0$, the solution obeys $$\begin{aligned} n(\cdot ,t)\rightarrow 0,\quad c(\cdot ,t)\rightarrow 0,\quad v(\cdot ,t)\rightarrow 0\quad \textrm{and} \quad u(\cdot ,t)\rightarrow 0\qquad \textrm{in} \ L^{\infty }(\Omega ) \end{aligned}$$ as $t\rightarrow \infty$.