Histograms are not useful for high dimensional data. The curse of dimensionality affects one quite fast. As in your case if the grid is of size 7**6, you have on average one point in one bin. Kernel density estimator are better suited as long as you keep the kernel bandwidth large enough. In my experience the top hat kernel as k-nearest neighbor yields reasonable results up to D=10, if sampling is sufficient. There is also a quite efficient algorithm for calculating k-nearest neighbors in higher dimensions, which I can recommend.
Also, the kernel shape doesn't really matter so much, because you need to keep the bandwidth large enough due to lack of data. If you see a dependency on the kernel shape your bandwidth is likely too small. There are a couple of rule of thumbs how to select the bandwidth.
If you calculate some other property from the probability density, in nearly all cases you are better off not computing the density at all.