My first-grade daughter has homework. Pretty much every night.
I still have a hard time getting my mind around this (I swear they didn't give homework till middle school -- or maybe late elementary school -- when I was a kid). But I'm not here to debate the wisdom of homework for 6- and 7-year-olds.
I'm here to say that I've tried to start teaching my daughter to proofread her work. It's that important. And you'd be surprised (or maybe you wouldn't) at the number of websites I have visited, presumably built and published by adults, that seem as though they haven't been proofread.
If your role includes posting anything to a website, proofreading should be part of your routine. Give it a second (or third or fourth) read before publishing, and look at it on the live site, too, just to make sure.
I'm a firm believer that readers notice spelling and grammar mistakes and judge you by them. Some might argue, in light of what texting has done to grammar and spelling, that fewer people care.
My reply: The few that do care are probably more likely to be the readers (or customers) that you want.
Before I go, a tip from my days proofing newspaper pages: Do one read for content -- making sure it all makes sense -- and one read for spelling. On that second read, I programmed my brain to stop at any proper noun and any word that was longer than about six letters to double-check its spelling. I have seen others recommend reading text backwards to check spelling. Makes sense to me.