The main downside is that it masks the problem and makes it less likely that you apply a better fix. If you are processing thousands of pages and don't want to hand correct that's a good thing, but if you are just doing a single book or thesis or something, it's better to see them and adjust by hand.
If the last line of a paragraph is just one or two words, and that short line goes over the page, then usually you can arrange the linebreaking of the paragraph not to have the final short line, for example
\looseness=-1 will ask TeX to make the paragraph one line shorter, or you can re-word the text.
If you do that then the paragraph fits on the page and the problem is gone.
If instead you just set the penalties to 10000, the linebreaking is not changed but TeX avoids breaking the page at the optimal place and so typically breaks it a line shorter, taking two lines over. This short page then has to be accommodated by stretching some white space somewhere.