Is spelling "old fashioned"? It better not be! Is caring about it old fashioned? That's probably more the question you're asking.
Firstly, if anyone gets upset over the question being asked, then I think that's a bit silly. It's a perfectly valid discussion topic. You didn't ask why someone can or cannot spell - you asked whether it was still important to think it is relevant to our modern life.
Spelling and correct grammar (not "grammer"

) are
THE backbone of our language. Without them we'll sink into linguistic anarchy, plain and simple. Correct practise and observation of these fundamentals are the only way we can reliably communicate with each other using a standard method. Yes, spelling changes over time as our language evolves to accomodate new events, inventions and the like, but in this case everyone should then endeavour to use the same accepted spelling for any particular word at that time. Technology should
improve our ability to communicate using this standard, not dilute it.
Those that can't spell, I can understand - there are quite a number of words to remember and we all forget from time to time. Being able to spell correctly the whole time is not a measure of intelligence - it is a measure of care. I look up my dictionary regularly as do most people who care about our language. Those that can't spell and really think it's unnecessary - I'm sorry, but I don't get your mindset.

Do you just not wish to communicate properly?
Grammar is just as, if not more, important than spelling. It's the way we use and place our words to define our message in a sentence to ensure its (note: no apostrophe in "its" in this case - we are not saying "it is" or "it has") context correctly. Without grammatical control the real message is lost and is then open to misinterpretation.
Try inserting some purposeful spelling mistakes in your next line of computer code some time, and see where that gets you if you think it's unimportant... LOL!

I guess that being in the advertising game, it's too important to get wrong. If we design a campaign and spell something incorrectly in a $six figure newspaper or magazine ad, it reflects badly on the customer. Chalk up one lost client....