6.D.

There are two types of such cross-sections:
(1) One side of the cross-section has one point, and the other side has three points, such as $\triangle B_{1} C_{1} D_{1}$ in Figure 4(a), where $B_{1}, C_{1}, D_{1}$ are the midpoints of edges $AB, AC, AD$ respectively. There are four such cross-sections, each with an area of $\frac{\sqrt{3}}{4}$;
(2) Both sides of the cross-section have two points each, such as the quadrilateral $MNPQ$ in Figure 4(b), where $M, N, P, Q$ are the midpoints of edges $AB, BC, CD, DA$ respectively. There are three such cross-sections, all of which are squares, each with an area of 1.
In summary, the total area of all cross-sections is $3+\sqrt{3}$.