"You bet they do.
You might not care a bit about an apostrophe that modify's the word incorrectly, but lot's of people care. They care because it is an instantaneous method for determining whether the person writing is facile with the language and/or cares about doing things correctly.
Now that first impressions are as quick as a few letters in a blog post title (sic!) or a sign in a window, every character matters."
Apostrophe misuse is rife in the UK. First impressions do matter and a little more care could so easily avoid giving the wrong impression. Bad apostrophe use says to me "I really don't care about getting things right". My grammar and spelling is far from perfect but I do care!