There are a lot of crisscrossing issues in this.
First off- businesses complain about "higher wages" pushing for things like more illegal workers that they don't have to pay shit, and to turn back the clock 100's of years of labor laws and conditions for. We in this country LOVE to willingly fall for this for some stupid reason.
So in many cases, businesses complaining about higher wages and the inability to find workers (ie: workers who will work for shit wages and feel they have very few rights or ability to improve conditions) are completely full of shit and just exploiting our nations (INSANE) tendency to allow the labor laws to be completely skirted in the name of some bullshit sense of civil rights for people who aren't even citizens here.
On the other hand- the push toward automation of things like fast food may work in some ways, but it'll fail in others. I personally don't eat much fast food, but when I do I want to order it from a human being. Pay that person whatever the market for their job will bear, but I will not order food from a machine. Places like Sonic here in California that have automated ordering systems I find confusing and not worth my time.
So maybe companies will save some money moving to kiosks, but they'll 100% lose customers like me. I have no idea which way will balance into more profits, nor do I care that much, it's up to those companies to figure out for themselves.
In other cases, I can see where artificially increased wages will hurt businesses. Just jacking around with numbers (including that of a minimum wage) does not lead to higher purchasing power in the long run, and that's the bottom line. Purchasing power trumps a jacked-with wage all day, every day.
If $15 becomes the new $5 in terms of what it will buy (and over time, I believe it will- minimum wage IS minimum wage- there's nothing that says it will remain a higher tier wage than... minimum over a long term) then nothing has been accomplished except to make a lot more people into minimum wage workers. Higher dollar amounts are also harder to come up with than lower ones in the long term. For example: I can probably walk around my house and dredge up $5 from under sofa cushions and such. It'd be better for that $5 to actually be WORTH MORE, than to expect I could just as easily dredge up $15 that only holds the same value eventually.
That same effect scaled across large businesses and swaths of the economy just means printing more money that will have less value and spending power over time. We should work to increase the value of each dollar- not jack with meaningless amounts of dollars. But of course that's a MUCH harder sell for the average idiot, who will always tend to think of $15 being better than $5... because MOAR!