It depends which Distro are you using? If it's a laptop I would set aside about 2 to 4 gbs swap for suspend and hibernate if you use them. If not with most distros today there is little need for a separate swap partition. One of the old rules of thumb was a swap 1/2 the size of installed ram. Some Distro use swap files in the root / partition as a swap area.
What KGill says. "Old PC" suggests "old rule" which was 2xRAM for swap space. My view is to let the kernel have the room to be as efficient as possible. The kernel is generally smarter than the user. Increasing the RAM size itself is a great benefit to the computer, so I'd do that if possible ... and still run 2xRAM if I could, though in the past I've increased RAM but left the swap space as it was because it was inconvenient to increase it.