Why you are not supporting IE6 and Windows 98 then?
- Currently and until April 11, 2017 only Windows Vista is the reason you might support < IE11 because IE11 is available only for Windows 7+. support.microsoft.com/en-us/help/13853/windows-li… There is no other reason why user should not have IE11. If he is lazy to upgrade it himself, then you should put a large screen with the link to Microsoft website providing instructions on how to do so or where to get Chrome/Firefox. In case of companies it would be administrator's job to install all the required software.
hurt the industry
Only the ignorance ans passiveness hurts any industry.
hurt the programmer
Allowing users to use unsupported software with bugs and security issues which will never be resolved will hurt business itself and your reputation also. This also would bring more problems and risk into dev team and code itself. So you actually hurting yourself.
doesn't help the customer
If you won't learn to say "NO" and won't force typical users to act, they will NEVER do it themselves. Only by acting you will help customers.
Microsoft not only offered a free upgrade to Windows 10 but forced everyone to do so. This is how you as a developer should act.
There are always alternatives available. Other browsers, other OS.
Believe me, companies, can buy and upgrade many licenses. Moreover I can tell you, they can use another browser. Bad companies and exec managers just want to save money everywhere even on own employees. However, even good companies will not act if you will not provide them all the critical information. It is your responsibility to say this to your boss/customer.
In many cases Microsoft provides discounts or even free licenses and services like startup programs, programs for schools, universities and students, etc. So "I can't have a Windows 10" is not an excuse.
Engineering is 20% of code and 80% of communication. Thankfully Microsoft did a lot of effort here, unfortunately, many small business can't do the same.
So, if it is not about Windows Vista, what is the reason why a specific customer you are talking about is not using IE11 or another browser? What direct answer they provided you?
At least you can use already existed infographics like this one which will explain to key business persons all the risks and that upgrading will actually at the end will be less expensive, the costs for development also will be less expensive (this is of course, if you respect your own time and add a additional fees on the top):
abacusnext.com/resources/infographics/risky-busin…