The width of the scrollbars depends on the Widget toolkit in use by your Desktop Environment not on the distro itself. This means that width is specified by the theme(GTK, QT, etc).
Because you are using Mint, I'm pretty certain it's GTK in your case, so changing the theme would be enough to influence it. Many sites have a great selection of them(just google "gtk themes" :D).
Also, you can manually override it by creating a file called
.gtkrc-2.0 in your home folder with the following line
X is the desired width in pixels.
EDIT: Here's a list of all the gtk properties of scrolllbars, it might come in hand. https://wiki.gnome.org/Attic/GnomeArt/Tutorials/GtkThemes/GtkScrollbar
EDIT2: Sorry, I didn't realize I forgot to mention that the that the option in
gtkrc-2.0 has to be associated with a widget. The correct way would be:
slider-width = 9 # for example
class "GtkScrollbar" style "scrollbar-style"
class "GtkVScrollbar" style "scrollbar-style"
class "GtkHScrollbar" style "scrollbar-style"
Similarly you can open your gtk theme file and search the same property and adjust it. Either way it requires a refresh for the latter and a logout for the former.