Back in my dating days I once Googled the names of potential future in-laws, and it turned up a record with a substantial and disturbing criminal history. The person was presently incarcerated. After asking the shadchan, my suspicions were confirmed that this was indeed the parent of the girl in question. I had mixed feelings about it because the girl herself was not at fault, but I wasn’t particularly interested in the shidduch to begin with, and the backdrop of having such an in-law was just too much for me to handle, so I said ‘no’. I felt really bad for the girl, but you shouldn’t get married for pity. Googling has proved very useful in avoiding such situations. I think it’s a responsible and prudent thing to do, and not at all unreasonable.
That being said, one should be careful to make sure to validate that the results they pull up indeed belong to the person they are looking into. I know that when Googling my uncommon name it pulls up some guy’s Facebook with photos of all his drinking, smoking, and clubbing activities… it’s not me.