No, the speed of your computer will only make the time it takes for the conversion to increase, it will have no effect on quality.
When using DVD Shrink, for better quality you should use 'Deep Analysis' before converting. For further adjustment, you can use 'AEC' (Adaptive Error Correction), which has 'sharp', 'smooth', etc. Max sharp will take the least amount of additional conversion time, while max smooth will take the longest. The descriptions are deceiving, as 'sharp' is not necessarily the best choice for sharpness. Basically, for less compression such as 75% or higher, use 'sharp'. For heavier compression such as below 70%, try 'smooth' or max smooth if it is very high such as 60% or lower. Also, you can adjust the compression for each title on the disc to give the best quality to whatever you want, such as applying less compression to the movie itself and heavier compression, still frames, or a still image for certain extras on the disc. You can also leave more space for the main movie by removing some of the audio tracks if you don't want them, such as foreign language tracks, director's comments, 5.1 or 2.0 track, etc. You can also remove the subtitles to give a little extra space.
Read the guides out there such as what was linked above and try making adjustments to each of the titles on the disc. If you don't care about the extras or menus, you can even backup the movie and nothing else, just read the guides.