How to Increase the thickness of the box lines in an R boxplot?

See the boxlwd parameter as discussed in ?bxp (linked to from ?boxplot). E.g.

boxplot(rnorm(100,50,10), horizontal = TRUE, notch = TRUE, boxlwd = 4)

Are you talking about the rectangle that surrounds the plot area? If so, then this can follow you plot call:

 box(lwd=5)