All common desktop browsers (Netscape/Mozilla, Internet Explorer, et al) in the last 10-13 years support HTTP/1.1. Internet Explorer 3.0 and Netscape 2.0 supported it as far back as at least 1996 (Wikipedia agrees with this).
Of particular note is that HTTP/1.1 is required for the HTTP HOST header, which is required if you want to serve different websites for different domains on a single IP address. As such if an HTTP client didn't support it, the user would be unable to access many websites.
I encountered an old smartphone browser that didn't support it about 9 years ago, but that's the last example I can think of where I'd even heard of an HTTP client that didn't.
I'd say that lack of client support for HTTP/1.1 does not need to be a real consideration today even if your doing something fairly esoteric (although I still remember it being a technical support headache over 10 years ago, trying to get people to upgrade from Netscape 1.x).
If you are writing a proxy you might want to think about it, but really even then I can't see it causing any real world problems as it's so fundamental to modern browsing. If someone is are running a browser as old (or as limited) as Mosaic 1.0 or Netscape 1.1 then they probably have bigger problems accessing content.
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