The "best way", would be to feed the Saturn's RGB into a high quality upscaler. Like the Framemeister or OSSC (framemeister's HDMI should be compatible with everything. OSSC might require some additional goodies like a DVDO device to normalize the signal for capture) Then take that HDMI signal and plug it into a good capture card, like the above mentioned Elgato or something else.
I use an older elgato game capture HD. Recording at 720p allows footage to be captured at 60fps. 540p and under it records @ 30fps. the elgato game capture HD allows direct connection to TV so you can play and watch from there. the pc/laptop will lag about a second behind when recording.
Had to purchase a separate s-video connection a year ago.
Saturn s-video capture
now I use RGB / OSSC / HDMI to gamecapture. When recording with OSSC it might be best to turn off the scaliness or reduce it as much as possible as it looks uneven through game capture, especially when the YT vids get scaled down for forum embedding.
RGB / OSSC (using the newer lighter scanline mode) / HDMI and edited in DaVinci