Remove strip background keep panel border

Setting panel.border might cover your graph unless you also add fill=NA to panel.border :

theme(panel.border = element_rect(fill = NA, colour = "black"))

One can also use:

theme(panel.background = element_rect(fill = NA, color = "black")

source (section "Panel Attributes")

If you set element_blank() for strip.background and keep element_rect(colour="black", fill = NA) for panel.border then top edge of panel.border will be black. As pointed out by @adrien, for panel.background fill should be set to NA to avoid covering of points (already set as default for theme_bw()).

ggplot(mtcars, aes(mpg, hp)) + geom_point() +
  facet_wrap(~carb, ncol = 3) + theme_bw() +
  theme(panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        strip.background = element_blank(),
        panel.border = element_rect(colour = "black", fill = NA))