The cheap ones usually have a problem with edges of the picture being in the wrong place, i.e. picture much smaller than screen (underscan), off-center, picture larger than screen (overscan), or any combination. Also, the picture clarity leaves a great deal to be desired.
The better ones cost more than a cheap TV set. I don't see the point.
If you're wanting to use your monitor because it's really nice, consider that the converted PS2 picture isn't gonna be. Don't worry about it.