There are many ways to do it. I can tell you how I do it. I bought a usb capture stick from Pinnacle installed the software that came with it and ran the video and audio lines to the pinnacle stick and captured it to the computer with the software. It cost about 50 bucks at the time when I bought it. If your computer is fast enough it will capture without dropping any frames. It only captures at 720x480 but since the laserdisc is less resolution than that anyway you aren't losing any quality.
For post-processing I use avisynth and avsp to clean noise from the image, splice the side changes, and most of the time I try to restore the video to progressive frames if possible. If I were you I would capture with the composite output...